The war wouldn't be world wide unless the apes could send them off themselves. It makes no sense on why they blame humanity only when it would take them or many countries to wipe it all out

We don't know what had started the nuclear war. Possibly another country (or organization) attempted to take advantage of the turmoil the ape revolution was causing.

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One would think that other countries would try to retake the U.S. unless they're in the same situation. Although it could make for a comedic parody to have someone with a obvious British or Canadian accident land near him with some futuristic technology, like a flying car, and indicate the astronauts had the bad luck to land in the "ape sanctuary."

Good question by the OP. Way back when ('70s and '80s) I'd always thought it would have been a great idea to explore for future installments of the "original" APES movie series. I felt there were a lot more interesting story possibilities than most gave this franchise credit for. Many of them were explored in comic book form.

it wasn't exclusive to North America, but since North America is the only country honest enough to continuously blame ourselves through our own movies, it does seem that way. in fact, it seems we're the worst country on earth simply because we allow ourselves through ourselves to be eternally scrutinized.
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I doubt the slavery was exclusive to North America. In Caesar's concluding speach he mentioned apes on five continents will be rebelling.

Maybe a little bit of both.