When does every event here happen?
Whenever would fit best. I was doing my best to come up with some ideas.
Miskatonic University and IOU Illuminati University just pop into exist, but once there due to time stuff they've always been there. They might have even been there before towns were founded around them. They are that strange and atemporal.
The Ashcroft Foundation is from the 2080s, but they would blend in rather well at any time. Drop them in around either 1999 or January 2038. That D-Engine is unlimited power. If you follow the directions, there is no issue. If you try to understand it or fiddle with it...
Wondertainment if they started as a catalog company they've been around since then 1960s. Folks always seemed to think they were an urban myth though. It wasn't until internet businesses took of in the early 90s though that they really hit it global.
Gensokyo's Great Hakurei Barrier falls around 2005.
Escafil Device is found around 1985. No real reason for that time.
The Kerbals crashed in 2000 nearby Hawaii.
The Sekirei dating game starts in 2016. Let's give them 15 years to get ready for it so the ship was found about 2001.
It's not known how long Doom Guy and his harem have been on Mars. They are discovered during the Mars is a Jungle phase. Turns out they are discovered during one of their bi-weekly orgies. Doom Guy is also worshipped as a sex god and is seen as the modern Chad.
Haruhi Suzumiya shows up around 2005 as well.
Camp Half Blood turns out to be rather recent so doesn't show up until 2020.
The Pandoravirus doesn't get discovered until 2070. That's an entire doomsday thing if they don't stop it.
London gets both steam ships and trains about 1925.
Beijing gets the MGE portal in 1870. It doesn't close until 1900 or so. A large percentage of China ends up having monster girl lineage.
Atlas gets invaded by the Gate about 5 years after they land in the Middle East. The gate doesn't close and is open and allowing trade and limited war in both directions. Atlas Aura users are kicking butt at the moment.
Tristain kingdom fits in better if its earlier. So they get put in about 1750.
Dojo of Ryozanpaku shows up in San Fransokyo about 5 years or so after San Fransokyo showed up.
Old Man Henderson was deaged and cursed young so his older self wouldn't be an issue. They didn't want him growing up in the same era either so he is 16 and landed in 1983. Because nothing could make him more boring than being a teen in the 80s. They also put him in West Texas.
Moperville actually showed up in the 1970s. It's generally a boring place though. So other than the magical users and gender flipping not much happens there.
Brakebills University ended up arriving in 1880. They've been in Texas for a long time.
Wallace Corporation HQ actually showed up right during the civil rights era of the 1960s. No one wanted mass-produced human-level robots running around being sold. They lasted about 2 weeks before going under.
Judge Dredd mega-block has been around and rather unchanged since 1870. No one can invade the place as it's very well defended. The locals don't bother leaving much. They don't try projecting force much. They do control about 5 miles out from the building though.
Griffin & Kryuger ended up arriving in 1950. This let Oscartown have glorious robot girls fighting for them.
The Moon Nazi fleet doesn't show up until 2020. They picked a very bad year to invade. They got hit hard by COVID.
The Constantines were banished from every version of Marvel and DC that they ever were on at once. Those settings finally got that fed up with them.. They arrived to London in the Trains having their bodies and souls being used as fuel for the transit. Hey, they are Constantines. That's seen as a minor starting punishment for them.
Elissa Megan Powers (Empowered), Ninjette, and Thug Boy get dropped in Gotham about 5 years or so after Gotham arrives. Gotham is rather well-patrolled at that point, but they are welcomed as another set of "heroes." O.k. Emp is a C lister at best and really doesn't belong anywhere near Gotham.
PS238 has been around since 1950. They actually do rather well for managing super brats and keeping them from causing issues.