[I realize the title of this topic can itself be construed as a spoiler; any mod is welcome to alter it as they see fit to fix it.]
Before getting started, it's important to point out that the Makoto Shinkai wiki cites Mr. Shinkai stating at the Grand Rex in France that Weathering with You takes place in 2021, and while the finale of Your Name takes place in 2022, the timelines are alternate and he doesn't care about the differences.
After watching Weathering with You and letting it simmer overnight, beyond a creeping growing appreciation I had for it, I was ultimately confused about the timelines of both it and Your Name. If they do indeed exist in the same universe (which, again, they don't), it makes for some conflicting timeline of events. Please refer to my 100% professionally made graph:
Review: Your Name. / Kimi No Na Wa.
In case it's too confusing, let me break down why these two movies cannot coexist, and what can be changed to ensure that they do.
(Note: When referring to Your Name, I define everything starting with the timeskip to be the ending, and the sequence when Taki wakes up, gets on the train, sees Mitsuha and meets on the stairs to be the finale.)
Post-timeskip Your Name takes place in 2021. It begins in October, where Taki searches for jobs and meets with old friends. However, the important bits take place in December, namely the rainy cafe scene and when Taki & Mitsuha briefly cross paths on the snowy bridge. The finale takes place in the spring of 2022, when they see each other from the train and finally meet on the stairs. While the ending is one long sequence, each cut (except in the finale) potentially signifies time passing, perhaps days or even weeks. [I believe the rainy cafe scene and snowy bridge scene are subtly implied to occur on different nights, due to the weather and change of inventory.] This is important for later!
Kimi No Na Wa (your Name) Explained!
Of course, Weathering with You takes place between July-August of 2021. In this time, we see Taki and Mitsuha living their separate lives in Tokyo, as they have yet to meet. It's also been the rainiest season Tokyo has ever seen, and as of August 22, has been raining nonstop for years.
The reason these movies therefore cannot coexist lies in Your Name's timeskip. In the ending sequence, namely October of 2021, no rain is seen. Furthermore, no rain is seen in the finale (Spring of 2022) when Taki and Mitsuha finally meet.
I believe a simple solution exists to cleanly merge the two movies with very little discrepancy and only minor changes to the timeline. I understand that retconning is herecy, however, if I had the ability to alter the events if only to make the transition between Your Name and Weathering with You seamless, here's what I'd do. (I'm only changing events in the ending of Your Name and leaving the timeline of Weathering with You untouched.)
Kimi No Na Wa.
First, push back the timeskip scene. We see adult Taki looking for work and meeting with friends before the summer of 2021, perhaps in the late spring. As all these scenes are dry, they simply need to happen before the big typhoon of summer 2021.
Second, push back the final few minutes of Your Name and completely remove all time cuts in the sequence. I made a big deal of mentioning that each cut signifies time passing; my simple solution is to get rid of that and align the endings of Your Name and Weathering with You.
The rainy cafe scene where Taki almost recognizes Sayaka and Tessie should happen hours prior to Taki and Mitsuha encountering each other on the snowy bridge, not days like the movie subtly implies. This would align with the scene in Weathering with You when the rain suddenly turns to snow due to Hina's mood. This also means that the cafe and bridge scene occurs the night of August 21, the night before Hina's birthday.
Trying To Explain The Timeline To First Time Viewers
The biggest change lies in how far the ending must be pushed back. Rather than occur in the spring of 2022, the ending of Your Name must occur the next day, the morning of August 22. This is simply to correlate with the one day of actual summer Tokyo got before the eternal rain. This is perhaps the biggest change because the scene itself would have to be tweaked; the cherry blossoms in the distance would be replaced with more seasonally adequate greenery, and the streets would be a lot more wet. Perhaps allude to the events of Weathering with You by adding puddles everywhere and whatnot.
Of course, my solution isn't a cure-all. There are problems with this, such as the actual train scene where Taki sees Mitsuha, as all the train lines shut down in Weathering with You. The only solution I can think of is a wave-of-the-hand; Tokyo, in a desperate move to get their logistically critical infrastructure up and running again, opened the least-damaged train lines in a matter of hours, and all passengers who needed to get somewhere boarded these trains as they could (Taki and Mitsuha included). Realistically, I don't know how plausible that is, but seeing as public transport is the biggest way to get around in Tokyo and is precise to a T, I wouldn't be surprised if they opened up what they could to relieve congestion and get the population moving again.
Another problem, more than just a logistical error, is how magical this August 22 is. It's Hina's birthday and the day Taki and Mitsuha meet again? Seems a little too fantastical. It's romantic at best and bad writing at worst, but seeing as fate and destiny are powerful motifs in Shinkai films, it isn't too far fetched to assume it's just an important day where lots of things happened.
Anime Your Name Kimi No Na Wa Mitsuha Miyamizu Facebook Cover
There are probably many other problems I'm not currently seeing, but I believe my solution does patch the major holes and allows the movies' timelines to converge as neatly as possible. I spent a lot of time mulling this over in my head, but I believe this to be the way to retcon the ending sequence of Your Name to coincide with the events of Weathering with You.
I don't think one can confirm that the two stories exist in the same universe, since the easter eggs of Taki and Mitsuha may be just simple cameos Shinkai decided to put in that aren't canon.
But now that you think about it, it's not too much of a stretch that when Taki and Mitsuha passed each other on the bridge and it was snowing, it was the same night when Hodoka and Hina ran away. Those events could've been the same night, which would line up perfectly with the next morning where it's sunny. On the sunny morning the next day, Hodoka goes to save Hina in the sky, and Taki and Mitsuha meet again for the first time in years. It would explain the unusual amount of puddles in the ending scene of Kimi No Na Wa where Mitsuha and Taki are running around trying to find each other. The streets aren't flooded of course, but I don't think it would make sense for Shinkai to include a random detail in Your name where the streets are flooded unless Shinkai already had the timeline for Tenki No Ko lined up with Kimi no na wa in his head when he was making the latter.
Anime Movie Review: Kimi No Na Wa, One Of The Best Anime Movies — Otaku Orbit
Not ALL of the streets were flooded the morning after Hina sacrificed herself, as we see when Natsume's riding around on the motorcycle, so it wouldn't be too unbelievable that the area near some staircase where Taki and Mitsuha meet isn't flooded either.