It might raise the chance of some conflicts (e.g. over Antarctica or Russia's border) and some land grabs, but I cannot see it leading to a world war.
So: you have two nuclear powers who are both relying on the same diminishing resource to feed their people. Do you not see how that could cause....tension?
I am not just some random internet person making this up. A lot has been written about this subject by respected researchers:
- https://www2.acom.ucar.edu/news/waccm-model-simulates-global...
- https://news.climate.columbia.edu/2020/03/16/even-limited-in...
- https://www.newscientist.com/article/2529589-a-nuclear-war-b...
- https://nsarchive.gwu.edu/briefing-book/india-and-pakistan-i...