Martian Manhunter used his telepathy to restore people's memories of the old timeline. The rest of it, I don't know.
An explanation that possibly works for most of it: in the new history, the 3 main earths (Arrow/Supergirl/Black Lightning) have all been one earth. However, in this history there were other earths and a revised version of the Crisis occurred that wiped out those other earths as far as they know. They are unaware that some were saved. So, Laurel and Nash are still from other earths, just earths that were destroyed in a post-Crisis version of Crisis as opposed to being re-written as never existing. And, it was this version of Crisis that has screwed up much of the physics.
However, Cisco should have both sets of memories (or multiple sets, as this is not the first time his past has been rewritten) and thus remember owning a Superman t-shirt. I think he should be freaking out more just from the existential crisis of there once being multiple Ciscos across the multi-verse and how does he know which one he really is? Who's to say that not more than just 3 earths were merged and he maybe a conglomeration of a bunch of different Ciscos. Should he think of himself as the Cisco from pre-Crisis or the one who has lived his whole life on the post-Crisis world?