I wouldn't expect that to change after a nuclear war, especially in a world without contraceptives.
That aside, there really needs to be more understanding of the difference between species extinction and cultural extinction.
Humans could survive for hundreds of millennia in a devolved state. They wouldn't be particularly human by our standards, but they'd have the same DNA.
But our culture would be gone. All the accumulated knowledge would be wiped out, and it's very possible nothing like it would ever return.
DNA is fairly robust. Culture, knowledge, and rationality are extremely fragile. We're not doing an outstandingly job of preserving them even without a nuclear war, and I can't imagine a nuclear war would make the situation any better.