The original simulations of nuclear winter back during the 1980s were probably overblown. The fact that some of the predictions for "extreme nuclear winter" sounded worse than the Chicxulub event, when involved several times more energy than every nuclear weapon that existed at the time going off at once in the same spot, should have really been a dead giveaway as far as that's concerned. The authors of the study even acknowledged that their results required a perfect storm of worst-case scenarios to be applicable.
With that being said, though, basically every study ever done that doesn't assume a best case scenario (ie., war occuring in the winter limits woodland wildfires, all other non-urban firestorms are minimal, and most particulate debris remains in the Troposphere where it can rain out) does predict some degree of cooling, along with damage to the ozone layer. It probably wouldn't mean the end of Western Civilization if India and Pakistan got into a nuclear exchange, but there's a good chance it would have global consequences.