You don't keep tanks around for what happens before the nuclear bombardment, you keep them around for afterwards. Most nuclear assets are pointed at either other nukes or military infrastructure; you'd still a large amount of civilian infrastructure around filled with angry civilians that would want to finish the fight one way or another before nuclear winter fucks everything. And tanks being tanks a fleet would survive everything short of nuclear carpet-bombing fairly well
I could be totally off base here but my impression, based on the staggering amount of nuclear warheads that could be deployed in a war, is that this line of thinking is wildly optimistic. I expect that any humans who survive a nuclear war will be focused primarily on staying alive and preventing human extinction, not continuing whatever geopolitical conflict led to the war.
