Not really.
Here’s what happened in the original movies: (Spoilers for the whole series, obviously)
Planet of the Apes: Charlton Heston and crew go on a deep space exploration mission. When they land they find themselves on a planet of apes. He’s trying to get away from the apes so that maybe he can get back on a ship and go back to earth. At the end of the movie, he finds a half buried Statue of Liberty. He realizes his module isn’t on another planet. It returned to earth. But earth has been taken over by apes who had overthrown their human overlords.
Beneath the Planet of the Apes: There’s still a society of humans on earth in the future. They fight with the apes that live on the surface. They live underground and worship the atom bomb. At the end of the movie faux Charton Heston is unsuccessful at preventing them from launching an attack to destroy the earth. Earth is destroyed by atom bombs.
Escape from the Planet of the Apes: In the third movie, it turns out before Nuclear apocalypse in the last movie, two ape scientists were able to escape using a time machine. They went back into the past to 1970s America. Americans received them well. They were celebrities. But when they find out that society in the future has been taken over by apes, they kill the two apes. However, it turns out the female ape had a child, Caesar. And Caesar is an ape taken care of by the circus. But he’s secretly a smart ape from the future!
Conquest of the Planet of the Apes: In the fourth movie, we see how there was a disease that killed off dogs and cats. So people started keeping apes as pets. And Caesar, the smart ape from the future was able to organize the apes and lead a revolt against humans who were mistreating the apes. And Caesar became the new leader of an ape society.
Battle for the Planet of the Apes: In the fifth movie, we see Caesar lead his ape army against the remnants of the humans who have started to go underground to survive. He wants to live in peace with the humans, but the remnants of the human race are pretty wretched. And by the end of the movie we see they have moved underground and have nuclear weapons.
So it’s a circular time travel story. It’s sort of a paradox. Smart apes from the future come back to the past. But even though it’s circular, it all ends the same way with a nuclear holocaust that destroys both and ape and man.
Pretty cool right? And it’s all internally consistent. Unlike Marky Mark’s version. If he ended up on a future earth like the Charlton Heaston original movie, then how the hell did he go in the past and discover an earth that had an ape Abraham Lincoln and ape police, etc? So it as an earth where it was apes all along and not humans? Some kind of parallel earth maybe?
Edit: Btw, this is all from the top of my head, and I haven’t watched the original movies since the mid-90s so apologies if I messed things up in my recollection.
Also if someone wants me to put that behind a spoiler tag, I can do that. But it’s been 50 years now since the original, so I think I’m okay.