While the first 14 sequels (with the exception of All Monsters Attack) do follow on in more or less the same world, because of the nature of the way films were viewed and distributed during the Showa 20's, 30's, and 40's, it isn't entirely accurate to describe them as having a continuity or cinematic universe in the same way people use those phrases today. The only way audiences or even the staff of the films themselves would be able to revisit a film is if it was being revived in theaters, was showing on television and happened to be in the right place at the right time, or there was a novelization, comic adaptation, or a super 8 available. Because of this, a lot of the finer details in between films that today would be heavily scrutinized were quickly forgotten, handwaved, or ignored altogether when creating a new story. Because of this, creating a consistent timeline connecting each of the first 15 Godzilla films as presented isn't strictly possible. In order to make all of them cohesive and fit together, some speculation and nudging is required.

The largest issue here is simply with how many Godzillas there are. Generally, the 2nd generation Godzilla is usually interpreted as being the adult Godzilla of all the films from Godzilla Raids Again through Terror of Mechagodzilla, which is technically possible, and that's how it's always been presented in various monster books over the decades. The standard interpretation of the various juvenile Godzillas, specifically the ones appearing in Son of Godzilla and Destroy All Monsters, is that they are the same individual. This is, of course, completely impossible, as the former states that World War II ended "20 years ago," meaning it must take place prior to 1975, while the later takes place in the 1990's. Nevermind how fast Godzillas may or may not grow, no animal on Earth doesn't age a single day in over 20 years. Except maybe jellyfish? But Godzilla is definitely not a jellyfish.

The question then becomes, if we are to make sense of this, what happened to the baby from Son? If it grew up, why do we never see two adult Godzillas in the later films? Popular fan theories usually suggest that the adult from the Champion Festival era is the adult version of the baby, which makes sense because the characterization is consistent, as opposed to jumping wildly from reluctant lesser of two evils to full on superhero actively seeking out space monsters. This would extend to DAM as well, but that leaves open the question of what happened to the 2nd generation Godzilla? Did he die? Another interpretation is that the baby died, which is easier to believe than an adult Godzilla dying off screen, but that makes the mismatch of the adult's characterization even worse. So you kill off Godzilla's only son and then he somehow becomes nicer? This is even worse if you consider the likeliest culprit behind the child's death was the induced freezing of Solgell Island... which was perpetrated by humans.

Because the films were never intended to be taken as a consistent universe, there simply isn't a correct answer to this or any of the smaller continuity issues between these films. So in order to undertake the exercise of making one, I'm going to have to pick a theory and stick with it. When it comes down to it, I find it much harder to believe that Godzilla's personality would tend towards being more Gamera-like after humans killed his son, than the possibility of him dying off screen. Both are hard pills to swallow, but if we assume the adult died, the most that assumes is that there is a story where that happens we just don't know, as opposed to the other option, where we are confronted with a scenario that just makes no logical sense. Maybe Destroyah appeared a few years early in this timeline? Maybe some other supermonster we've never seen did it? Maybe he got diabetes? Basically, I'd rather the answer to the question have to be filled with fan fiction than with undoing 7 movies of a consistent character arc. That doesn't mean my interpretation is correct, or that I've even thought of every possibility, of course, it's just my interpretation. You're welcome to your own.

Key

Superscript = These are the number of the film in the series where the information comes from, with 1 being Godzilla, 2 being Godzilla Raids Again, etc. Additional superscripts will indicate supplementary sources not present in the films themselves, which will be explained at the bottom of the page.
Bold = Below the year heading, dates and times in bold are ones that taken directly from the films. These are anchoring points used to figure out when the rest of the events take place.
Alert Level 1: Issued when any scientific, geological, meteorological, or psychological evidence, except for the physical evidence, of G's activity has been confirmed.
Alert Level 2: Issued with G's physical activity, such as voice and movement, has been confirmed.
Alert Level 3: Issued when G has appeared.
Alert Level 4: Issued when G would certainly land on a specific site in Japan.

251,939,051 to 10,008bce

3036bce

The space monster King Ghidorah arrives on the planet Venus to destroy it an drain the life from it. It encounters opposition from an advanced civilization there. King Ghidorah ultimately succeeds in wiping out all life on the planet, leaving only a few Venusians left to flee to the Himalayan Mountains on Earth.5

1944

February 6th, 21:00 - A US battleship stationed off the shore of Lagos Island begins shelling the island attempting to flush out a Japanese garrison stationed there.18
February 7th - US infantry invade the island and clash with the Japanese garrison led by Major Yasuaki Shindo. During the skirmish, a Godzillasaurus living on the island is disturbed, and lashes out at the troops. The US troops fire back at the dinosaur, but it wipes out the landing forces. It is injured by the US battleship firing on it and retreats back into the jungle, and the US decides to withdraw from the island.18
February 14th, 18:00 - Major Shindo's garrison travels into the jungle to salute the injured Godzillasaurus before boarding a ship to return to Japan.18

1954

March 1st, 6:45 - The US nuclear test Castle Bravo is detonated at Bikini Atoll in the Marshall Islands. The blast is more than twice as large as predicted at 15 megatonnes, making it the largest nuclear detonation ever unleashed by the country, and was the largest in history at the time. Fallout from the explosion fell for miles around, well outside the predicted range, causing contamination of fish and people living in nearby islands. The Japanese fishing boat Lucky Dragon No. 5 was among the victims, killing one of the crew, and sparking renewed anti-nuclear sentiment in Japan. The bomb also mutates a Godzillasaurus that was in the area into the 1st generation Godzilla.
August 13th, 19:05 - The Eiko-Maru, a 7,500t freighter belonging to South Seas Shipping, loses contact at 24°N 141°2'E after being sunk by Godzilla.1
August 14th, 17:00 - Hideoto Ogata, a diver for Souther Seas Salvage, cancels a date with Emiko Yamane due to the disappearance.1
2:35 - The Bingo Maru arrives at the same location but is also sunk.1
August 15th - A helicopter and two more ships are sent out, the Kozu and Shikine11. Meanwhile, three survivors are found by a fishing boat from Odo Island, who describe the sea "exploding." The fishing boat carrying the survivors heads back towards Odo Island but is sunk before it can get there, making three in total.1
August 16th, 18:00 - Another boat, this one owned by the Odo Islander Magobei, is sunk.11
19:00 - Masaji Yamada, a survivor from the fishing boat, is discovered floating unconscious off the coast of Odo Island and brought in1.
August 17th - The fishing boats of Odo Island return to shore empty handed. The reporter Hagiwara travels to Odo Island and interviews Masaji and a village elder. The elder tells him of the island's legendary monster Godzilla, that in the past there were sacrifices used to appease the monster when fishing was bad, and Masaji says the reason for the sinking ships is a giant creature. That night, said legendary creature comes ashore on Odo Island and goes through the village, killing Masaji and his mother in the process, leaving Masaji's brother Shinkichi an orphan.1
August 18th - A group from Odo Island gives a report National Diet, including Hagiwara. In addition to the losses, they report the appearance of the monster. The paleontologist Dr. Kyohei Yamane gives closing remarks where he intends to head to the island to conduct research.1
August 19th - The investigation of Odo Island discovers radioactive contamination, a trilobite, giant footprints, and finally Godzilla himself, peering over a mountain.1
August 20th - Dr. Yamane debriefs the National Diet on his hypothesis of Godzilla's origin. He suggests the creature is a type of Jurassic marine animal that partially re-adapted to land during the Cretaceous period. Further, Godzilla seems to be adapted to deep sea environments based on sediment samples, explaining why it has been undocumented by science until now. The presence of strontium-90 in soil samples as well as radioactivity in Godzilla's wake directly ties his appearance to nuclear testing. This sparks a fight among the Diet about whether to make the information public, due to concerns over panic and potential damage to foreign relations.1
August 20th to 25th - Dr. Yamane adopts the orphaned Shinkichi.22
August 25th - The Disaster Response Center is established.1
August 26th, 4:00 - Godzilla sinks the 530t oil tanker Sakae Maru, the 17th such ship attacked.11
10:00 - A fleet of 10 Tacoma-class patrol frigates leave port.1
10:17 - The fleet reaches the designated area between 138° to 138°7'E and 33° to 33°8'N and begins dropping depth charges, attempting to destroy Godzilla.1
20:00 - Passengers aboard the pleasure cruise ship Tachibana Maru witness Godzilla rising from Tokyo Bay for a moment before swimming away.1
August 27th - The Disaster Response Center calls on Dr. Yamane for advice in dealing with Godzilla, but not only does he not have any advice on how to deal with a monster that's already survived a hydrogen bomb, but he disagrees with attempts to kill Godzilla in the first place, pointing out how valuable he is as a specimen for study given his ability to do so.1
August 28th - As the new paper comes off the presses containing the details of Dr. Yamanae's statements, Hagiwara's editor asks him to seek out an interview with Dr. Daisuke Serizawa, a colleague of Dr. Yamane who is engaged to his daughter, Emiko. He is initially turned away, and so asks Emiko to help him, which she does, but he doesn't get anywhere. Afterwards, Dr. Serizawa shows Emiko what he's currently working on under the promise that she not share it with anyone. This leads to a demonstration of the Oxygen Destroyer, which vaporizes a tank full of fish in moments. That night, Godzilla surfaces in Tokyo Bay and briefly rampages through Shinagawa before heading back out to sea.1
August 29th - The morning after, a meeting with researchers from around the world devise a plan to build an enormous electrical barrier, 30m high and 50m deep, around the perimeter of Tokyo Bay.1
August 30th to September 20th - The electric barrier is constructed.1
September 20th, 16:30 - Godzilla is spotted 15 miles northeast of Kannonzaki headed northwest.1
17:40 - Godzilla is spotted out at sea changing course northeast, zigzaging up the bay.11 Residents of Minato, Shinagawa, and Ota wards are ordered to evacuate.1
19:30 - Ogata makes a series of shitty decisions just before Godzilla surfaces. Godzilla walks through the barrier, melting towers with his atomic breath, and moves onto the rest of the city. The JSDF assault the monster with battalions of tanks and jets, but none of them make a scratch. Godzilla demolishes a large portion of the city, leaving Minato, Chuo, and Chiyoda wards a sea of flames. The death toll is staggering and those that survive the destruction are contaminated by Godzilla's radiation.1 Among the victims of Godzilla's rampage are the parents of Makoto Hayashida.16
September 21st - While volunteering to help the victims in the aftermath, Emiko is horrified after seeing firsthand the toll Godzilla has taken. She is convinced to break her silence on Dr. Serizawa's Oxygen Destroyer when she realizes it may be the only thing that can stop Godzilla. After telling Ogata, the two of them confront Dr. Serizawa, and eventually convince him to use the device as a weapon against Godzilla. The only condition is that it will never be used again.1
September 22nd - With everything in order, Ogata, Emiko, Dr. Yamane, and Dr. Serizawa sail out on the Shikine Maru to Godzilla's resting place, out in Tokyo Bay midway between Kawasaki and Kisarazu.1, 22 Ogata and Dr. Serizawa dive down to Godzilla and Dr. Serizawa activates the Oxygen Destroyer, but once he does he cuts his cord, preventing those topside from pulling him back up. In this way he sacrifices himself to ensure that the Oxygen Destroyer can never be used again, taking Godzilla with him.1

Sources

Times for sunset, sunrise, and the like use the calculations from the website SunCalc.

1 - Godzilla. The time of Eiko Maru's sinking is given right at the start. We're never explicitly told it's 1954, but the movie is clearly contemporary, not just from the fashion, the look of Tokyo, the vehicles, etc., but also from the fact that this movie is a direct response to the Lucky Dragon No. 5 incident and directly references it in the dialogue. So it's at least August 1954 at the earliest, and it can't be August 1955 because newspapers in Raids Again show us January of Showa 30. Not that this is in doubt, but if you for whatever reason needed more proof, the calendars in Ogata's office match up with August of 1954, and perhaps most glaringly obvious, the entire rest of the totality of the Godzilla multimedia franchise puts the events of this film in 1954. After a clear time jump we're given the times for the depth charge assault, but not the date, and after another time jump we're told Godzilla is sighted at 16:30, but again not the date.

11 - The first novelization, which is actually based on the radio drama that played before the theatrical release, combined with an earlier draft of the script. The info here comes from a fan translation of the novel which you can read for yourself here. All of the dates and times that appear in the finished film are the same in this version, but there are additional times as well. Whether these extra details are from Shigeru's story, the draft, or the radio drama I can't say, but they were very helpful in figuring out the length of the two time skips.

In this version, the day of Hagiwara's interview with Dr. Serizawa, him showing Emiko the Oxygen Destroyer, and Godzilla's first landfall in Tokyo is called a Saturday. Because there is a clear time jump between the Diet fight and the headlines noting that the number of sunken ships has risen to 17(!), and because the 20th was a Friday, it must be later than the 21st. While this puts the day where Godzilla spooks the party boat (named the Tachibana Maru here) on a Wednesday night, which you would think would be a thing you'd save for the weekend, there's really no other interpretation here that makes sense. The papers about Dr. Yamane's statements are brand new on the 28th, so they happened the previous day, and the reason he's even being talked to in the first place is because of the reaction to Godzilla's appearance.

The other big time skip in the film is during the period where the electric barrier is being built. In the original story this was a more protracted event, with their being tension related to whether the barrier would be built in time for Godzilla's next appearance, or even whether he would appear again at all and the whole thing wasn't just be a waste of money. I remember a commentary track talking about this before and I'm fairly certain the time span they gave was something like two weeks. This version does help, but it's not as helpful as I would have liked, because it merely mentions it being "day 20," and I don't know if that means it's the 20th day of construction or the 20th of the month, or something else. Maybe it's a translation error? At any rate, if the soonest they could start construction is the 30th, then the 20th day would be September 18th anwyas, so I went with 20th because "20th day of the month" sounds far more natural to me than "20th day of construction" without context and the two are so close it barely matters.

5 - Ghidorah, the Three Headed Monster. Princess Salno's Venusian memories tell us King Ghidorah appeared on Venus 5,000 years ago. This is obviously meant to be approximate, but 5,000 years before 1965 is astronomical year -3035, or 3036bce, or 4985BP if you wanna get real fancy.

18 - Godzilla vs. King Ghidorah. The dates here for the events on Lagos Island come from the onboard computer of the time machine KIDS. Of course, in this timeline, KIDS never traveled back to 1944, but the events that don't include it still occurred.

22 - Godzilla vs. Destroyah. It's made explicit here that the reason Shinkichi hangs around the Yamane household is because he was adopted by Dr. Yamane after he was orphaned. Incidents in the construction site of the Tokyo Bay Aqua Line are said to happen in the same spot the original Godzilla was killed, making it between Kawasaki and Kisarazu.