No, speaking about the original timeline, the Nuclear War happened at the end of the 20th century - as established by Cornelius' research in POTA '68. Thus, even if you don't consider the Alpha/Omega Bomb as "canonical" within the background and the intentions conceived by the writers of POTA '68, the Nuclear War matches what we see in BATTLE. Caesar wasn't the "spark" of the Ape Revolution in the original timeline. The Ape Revolution occurred in a huge timespan of 300 years. 1993 + 500 = 2493. 1485 years before the arrival of Taylor. Apes never "mutated". You cannot make normal apes evolve into Humanoid Apes in few years, that would be IMPOSSIBLE even from a sci-fi standpoint. As simple as that, Humanoid Apes already existed in that universe, and they came from Africa and Asia. They were there from the start. I guess that, in the original timeline, they had been brought to America in the 1973-1993 timespan, not for slavery. Remember, in the original timeline, Cornelius, Zira and Milo never arrived.