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Generally, it’s unsure if nuclear winter would actually happen or not. We would surely see global cooling/temperature dips as we saw that with the oil fields in Kuwait so if you mean just a big global cool, yes. Otherwise I think a lot of scientists agree that rain and weather would bring down a lot of the soot and shit in the atmosphere.

Bigger problem in my opinion would be whatever is getting vaporized in major cities and military facilities getting sent into the air. Power infrastructure would likely be targeted as well e.g nuclear power plants. Chemical warehouses/storage facilities. Major industry. All of that would be a chemical nightmare that would be compounded by getting burnt/vaporize and soaking into the environment. I mean, anti-fire foam is horrible if vaporized for example.

“Whoops!”