I was recently re-reading World's Finest #81, where Superman and Batman are visited by a historian from 5956 who wants to confirm that the history book he has written about them is accurate (it isn't, unsurprisingly. Just imagine if you took a history book about the New Kingdom of Egypt back to the reign of Ramases the Great.) Anyway, he mentions in the story that the true identities of Superman and Batman were not revealed until long after the present day.

This got me wondering, when was Batman's true identity reveal in the future of Earth-1 and more broadly, what is his legacy in the future? Note that not all these events may take place in the same future, but for the sake of completeness, I will mention them all.

According to the Hex timeline, a historian named Cohen will deduce Batman's true identity in the mid-21st century, and use that knowledge to rediscover the Batcave and become the new Batman. Presumably, he didn't publish his discovery.

In 2467, the currently Batman is named Bron Wayn. He is the 20th Batman, and confirmed to be the direct descendant of Bruce Wayne. His father was the 19th Batman, before being killed by the 19th Joker (also directly descended from the original Joker) He fights the 20th Joker once, but then that Joker dies in battle before he can reveal the hiding place of his retired father. By this point, the Wayne family live on a private asteroid, where the current Batcave is.

by 2693, Batman is regarded as a legendary figure, with only a minority of historians believing that he really existed (and even they are unsure if there was ever a Robin). Tomas Wayne is the leader of the Batmaniacs, a society of Batman enthusiasts, though he does not know that Batman was Bruce Wayne. That knowledge seems to be lost

In Adventure Comics #341, the Legion of Super-Heroes re-discover the Batcave (lost for centuries) and hide out there from Computo for a while. Brainiac 5 finds in the Batcave the Bizarro ray projector (no idea why it is there and not in the Fortress of Solitude) and a device used to imprison an anti-matter monster, which the Legion uses to destroy Computo. By this point, Wayne Manor is totally gone. There is only an open field (maybe a park) over-top of the Cave.

In Brave and The Bold #179, Batman visits the 30th century, and meets the Legion of Super-Heroes. They recognize him as a hero of the 20th century named Batman (after a bit of thought, they weren't quite sure of the name). It is mentioned in the story that Bruce has a descendant named James Gordon-Wayne, who is a famous detective and whose arch-enemy is a criminal named Halkor. However, the Legion do not seem to be aware that Bruce Wayne is Batman. James Gorden-Wayne may be the same as the private detective who is killed by Professor Ivo in Legion of Super-Heroes Annual #1. However, Brainiac 5 mentions that he knew that detective was descended from Batman, which suggests he found out Batman was Bruce Wayne in-between the stories (possibly the 30th century historians use the mass availability of time-travel in that era to solve many mysteries of history).

There is no evidence on Earth-1 that Brane Taylor (Batman of the 31st century) is related to Bruce Wayne. However, his Earth-2 counterpart Brane clearly is, so it is possible. He was inspired to become Batman by reading historic records, and if he is related to the original, he doesn't seem to be aware of it. He knows well that Bruce Wayne is Batman, and it seems to be common knowledge.