Nuclear winter came later. During late 50s early 60s (my high school and college years) ice age was all the talk. Lots of articles in credible popular magazines like Readers' Digest, Saturday Evening Post, Time, Newsweek, etc. Those articles all cited scientists in scientific journals and scientific conventions. But it was main stream science that originated the data and analysis. No internet, the public wasn't as much in direct contact with scientific papers as they can be now.
The thrust was purely analyzing trends in recent data and it was viewed as a natural phenomena. Thus no politics. Just recognizing it's happened before it can happen again.
History of the nuclear aspect here.
There was nuclear work dating to the 50s coming out of the current nuclear development and open air testing but Sagan et al didn't capture the public's attention until 1982. I believe it was the Sagan group that coined 'Nuclear winter'.