Actually, up-to-date modelling suggests that even a "minor" nuclear war between only two combatants, with 50 warheads apiece would be enough to render global agriculture impossible for a year or longer. The concomitant effects on hunter-gatherers are probably similarly devastating.
If some portion of humanity does survive that first year, I wouldn't be so very optimistic they're close enough to each other to make rebuilding a minimal viable population easy, let alone that the memories of the recently-destroyed global infrastructure are sufficiently present and relevant to be worth carrying forward to their descendants as anything other than a cautionary tale. What you're looking at is basically a remote chance that some really isolated group in say, the Far North or an island in the Pacific manages to hold it together in a hunter-gatherer kinda way and do so for long enough that their population doesn't collapse.
I'm betting you still don't see them expand in any meaningful way for centuries after the fact, and the environmental damage may constrain even that for a lot longer.
Either clarification or citation humbly requested. The United States alone has conducted over 1000 nuclear tests, without rendering global agriculture impossible.
A reminder for everyone: on this day in 1983, Stanislav Petrov saved the world.
It occurs to me this time around that there's an interesting relationship here - 9/26 is forgotten, while 9/11 is remembered. Do something charitable, and not patriotic, sometime today.