Yes, I'm aware that humans were nomadic hunter gatherers back then, and for most of our history. Sure, any small change can be catastrophic for such a culture. Now, having seven billion humans depending on an intact ecosystem would be a bit of a problem as well.
I most certainly hope that your assessment of the dangers of nuclear war and radioactive fallout are accurate. If they are, it may not be necessary to rebuild civilization over thousands of years at all. Maybe such a war would actually wake up the human race to the follies of war.... oh, wait. That was what WWI was supposed to have done.
It is all involved with the half-life of the particles created. Remember, the vast majority of fallout is not in itself radioactive, it has become irradiated. And the half-life of these particles is very small.
The rule of thumb is the "7-10 rule". That is for every factor of 7 hours that pass, the amount of fallout that has lost it's radiation is increased by a factor of 10. So within 2 weeks it has reduced by a factor of 1,000. The only particles that are really a concern are those form the remaining fissile material (uranium and plutonium), and a few other elements made in a nuclear reaction, like strontium 90 and caesium 137. These are basically "leftover" particles from the fission of uranium and plutonium, and have much longer half-lives. Roughly 30 years for each.
And even then, it has to be realized that the majority of this will be left in the immediate downwind range of the blast. Most of the rest will end up in the sea in relatively short order. It is essentially dust after all, and is easily washed away (that is why water is the primary means of decontamination).
The only real long-term risk is in the consumption of concentrated fallout. In other words, meat. One of the dangers in any contamination is in the consumption of animals that eat lower on the food chain. So if cows are consuming a large amount of plants that are contaminated, they will in turn concentrate it in their meat and milk. This is why we avoid things like fish and shellfish from areas contaminated by sewage or other urban runoff.
So want to avoid this? In the first year or so afterwards try to eat as many well washed plants as you can from outside of the immediate fallout zone, and restrict your consumption of meat.
Remember, it has been 72 years since the blasts at Hiroshima and Nagasaki. And outside of the immediate blast area there was never any real steps taken to decontaminate the area. And the radioactivity there was down to the normal background levels within less than 30 years. And of those born after the war, the cancer rates are no different than the rest of Japan (which are among the lowest in the world).
And unlike vulcanism, it would be a one-time event. Bang, and it is over. So at most, a repeat of 1816 after Mount Tambora blew up. Known as "The Year Without A Summer", there was snowfall in Albany, New York in June. But that was also the largest of 6 separate eruptions over a 6 year time frame, so the effects snowballed each year.
But it always must be remembered that for every cause there is an effect. The failure of much of the New England crops that year caused a migration to the Midwest, specifically Illinois and Indiana. It also caused Mary Shelly, Lord Byron, and several of their friends to hang out at Byron's Swiss chalet and tell "ghost stories". Which gave us both Frankenstein, as well as an unfinished story by Byron that was finished by another guest at that months long party, "The Vampyre". This is what started the "vampire craze" in English literature, that culminated in Dracula.