As far as I understand history of this claim, back during the Cold War it was common to predict that even small a scale nuclear exchange will get the world back to long term Ice Age, due to widespread urban fires. Similar predictions were even made about Kuwait oil wells fires in 1991 (it's a good model, as the effect was not supposed to be related to nuclear explosions as such, just resulting fires).

It turns out from more recent models, and actual data from the Gulf War that the actual magnitude of cooling is orders of magnitude smaller than what was predicted, and there was never any genuine research that really suggested levels that were widely claimed; the most straightforward explanation is that people opposed to nuclear weapons wanted to exaggerate its effect to scare people off. It might have been a calculated lie, or something they genuinely wanted to believe - the point is that politicized science is not very accurate even if you agree with its political goals.