Well, those who have studied even a limited nuclear war have argued that we can expect serious climate change resulting even from a limited nuclear war, on another continent. So if the war is small- a few nuclear strike on a limited area, perhaps one could see little global change. But it doesn't seem to take a lot to start causing serious systemic challenges. A major if limited nuclear war between Pakistan and India would have serious consequences climatically. I would think a major nuclear war between Israel and Iran, would even be more dramatic, and of course Russia vs China could have huge systemic consequences. In none of these cases would the "the West" be directly impacted (except perhaps through ancillary fallout impacts on South Korean and Japan which might include in the developed, democratic West).
We live in a globalized world- and that's not just economically globalized (the financial consequences of any of the possible wars mentioned above would be significant on the global economy) by environmentally globalized. Increases in Ozone depletion, climate change conditions would, depending on the size of the war, have systemic consequences.
If we are talking about a Fallout type war, than I think there are two possibilities. Perhaps I am one of those lucky ones who has found shelter in a former mine turned fallout shelter, where there is 1 man and 10 women and our goal is to repopulate the human race. If that's the case, than I say, "Turn on the blender, grab some limes, mix some drinks, and let the party roll." If I am on the surface... well, I live next to a military base so I am probably going to put on the shades, a heavy sunscreen and expect one hell of a burn coming.
If you are living in the US, you might want to think about the following maps for nuclear targets, prevailing winds and places were you have the "best chance" to survive-
http://modernsurvivalblog.com/nuclear/us-nuclear-target-map/
I should also add that much of our "apocalyptic" thinking suggests a world of violence and chaos driven by two elements (1) lawlessness and disorder and (2) scarcity. In short, this is a world that looks a bit like the poorest developing countries- Somalia for much of the last 20 years. But this might not make a lot of sense.
Let's assume however, that scarcity is less a problem- depopulation has left lots of resources abandoned, food supplies in department stores, land for agriculture, solar panels, energy resources. So the scarcity hypothesis might not fit- except for one thing- people. One needs people to survive because humans are social creatures. I would assume in a nuclear war- a lot of people suddenly dead and a lot of infrastructure devestation. But if the war is more limited nuclear but much more biological or chemical- so we have a loss of human life but not infrastructure- we may have a different level of scarcity. SO what about disorder? Well the "lawlessness" problem suggests a Hobbesian world of man at war with man- and life is "nasty, brutish and short." Except, Hobbes points out that man overcomes this through government. Given sparse populations and the need for cooperation, I think the lawlessness problem would be overcome. There is more to be gain through cooperation and trust than distrust and competition.