Most current estimates are that Mars' cataclysm was 80-100 million years ago. ...which means they really have no goddamn clue, though.
It's entirely possible there was no Mars Nuclear War and the planet was actually annihilated by the same thing that killed the dinosaurs. Probably the explosion of Planet 5. Sent pieces of it flying all over the place. Mars got pummeled, Earth took a couple big hits and that was enough to get rid of the dinosaurs. This would put the explosion of Planet 5, and the destruction of Mars, at about 65M years ago (since our ability to date the extinction of dinosaurs far exceeds our ability to date Mars hypotheticals).
The biggest issue with "we're from Mars and they had a nuclear war" is that there seems to be a vast gulf of time between Mars going tits up and modern humans appearing on Earth. Even if we're going at the low end of scientific estimates of 80M years ago (or my own guess of 65M) that's still tens upon tens of millions of years between that event and modern humans appearing 200k years ago. What did they do for 64.8M years?
Another big question for me: What blew up Planet 5, anyway?