If nuclear war had happened in 1980 or later instead of 1962, would it lead to a full-blown Ice Age due to the greater number of targets and greater accuracy in hitting those targets (creating more dust)?
I'm not a climate scientist, so I would defer to more informed opinions but I think it is clear that there would be a climate effect of cooling but the intensity and duration of that effect is an open question at least in my own mind. By the 1980s, though, perhaps the bigger question is whether there would be many people around to care. Europe, North America, the Soviet Union, Eastern Asia and China would be devastated. A nuclear war in 1962 would have been bad. By 1980, it became unthinkable. If anyone was around to care, a cooling climate effect would complicate recovery by creating a harsher climate and by making the growing season shorter. If you look at that map, by the way, the most irradiated parts of the US happen to coincide with some of the best farm land in the world. Those who didn't die in the blasts or from the fallout would face a very difficult food situation very quickly. And those in other parts of the world dependent upon food imports who may not have been directly affected by the nuclear blasts and fallout would feel the impact rather quickly. And climate change, even temporary, would make that worse.
While I don't think nuclear war even in the 1980s would have killed off humanity, it would have been a mass extinction event reducing the human population from billions to millions, with the survivors facing a long road back to an industrial civilization. And a cooling of the climate would have played a role in that, whether it lasted a few years or turned into something longer.
Fortunately, we didn't blow up the world, so we can look back on this 30 years later and discuss it! But it was certainly an odd era in which to grow up and contemplate such questions.