I'm wondering if a better way to go about this would be some biological agent, rather than nuclear. If a super-virulent disease were to be created that could wipe out 99% of the population and drastically reduce the life span of the survivors, then whoever is left will be scraping by for survival. They wouldn't have time to spend teaching the younger generation anything other than the basics of how to survive, which can lead to a loss of knowledge… but that's till quite a stretch.

A nuclear winter may result in something similar, but it would have to be something rather big. The Krakatoa eruption caused global cooling and a reduction of crop production worldwide… for about a year or two. The asteroid impact at the end of the Cretaceous, on the other hand, caused cooling for long enough to kill off the dinosaurs, but that required an asteroid 10 km across to impact the earth. This would be the equivalent of 100 teratonnes of TNT, whereas the entire worldwide stockpile has a yield of 5 gigatons, 1/2000th of that of the yield of said asteroid…