Because it's time to go to war https://www.huffpost.com/entry/nuclear-war-global-warming_n_...
And likewise, not sure it would favor the United States all that much either since 1/8th of our landmass is in Alaska. It'd probably be a wash for the US but certainly Russia has no reason to use nuclear weapons to halt or reverse global warming.
I could see this conspiracy theory being slightly more believable for an exchange between India and Pakistan, though. And by slightly, I mean only mildly less laughable.
EB has a surprisingly short and to-the-point summary of it. https://www.britannica.com/science/nuclear-winter