I just realized something that might be obvious to more experienced DC readers. I found the latest volume (#8) of the New Teen Titans trade paperback collection at the library, and toward the end of it, there was a bit where one of the villains the Titans were fighting suddenly got teleported away by the Monitor, who told him the CRISIS ON INFINITE EARTHS was coming (with the actual logo in his speech balloon) and his help was needed. So I looked into it more, and apparently the Monitor went around recruiting heroes and villains from across the multiverse to prepare for the Crisis.
So that seems like the simplest explanation for how all these heroes in "Elseworlds" are brought together -- the Monitor recruits them all to help with some crisis, though not necessarily the Crisis, which is still supposed to be 6 years away (although all the timeline alterations the speedsters and Legends have brought about might change that). And since he can recruit multiple heroes from a single world (e.g. Flash and Green Arrow), that doesn't rule out Batwoman being from Earth-1 or Earth-38. Anyway, it's possible that the scenes in Gotham City or the stuff with Clark and Lois might be setup scenes, showing the characters going about their normal lives and adventures before the Monitor whisks them away.