I know there would probably be an ash-based cooling period, but, how long would that last and then what happens? Is the post-apocalyptic desert even possible in a post-nuclear war world?
We don't know.
There will be several factors which affect the climate in different ways. There is no empirical data on which will dominate.
- How many weapons and where? Arsenals are much smaller than they were at the height of the Cold War, and we can assume that Moscow, Washington, and places like Minot AFB are still getting overkill.
- Firestorms could blow ash into the higher atmosphere and cause a nuclear winter. That could be just a brief effect, a few years or a decade, or it could lead to increased snow/ice cover and increased albedo. That lowers the temperatures further, and a feedback loop occurs.
- Or the firestorms could lead to deforestation in some fragile ecosystems, and runaway desertification.
It could also be that the death of many humans reduces deforestation, instead.
- In addition to direct climate effects, killing people and economies will also matter, but 200 years could be enough to get people back on their feet and emitting greenhouse gases as before.
Imagine you have a bucket full of water, with a couple of holes in it that cause a steady loss of water. There is also a water pipe above with slow leaks, steadily refilling it. The water level in the bucket is mostly stable. That's our climate. And then somebody gives the whole Rube Goldberg contraption a good shake. Nobody can predict if the water level will rise and drop after that. Only that the shake is a very bad idea ...