Unfortunately I'm not sure that our politicians self-preservation instincts, strong as they may be, trump their egos (sorry for the pun.)
As soon as you launch that many weapons, and kill such a high percentage of people, you eliminate the possibility of manufacturing new weapons, so the war would be instantly over.
And I believe that despite others simulations to the contrary, there will be small groups of people who will stay underground for years, surviving on tinned etc. food. After 5-10 years, radiation levels will still very high, but most dust will have settled, and the skies clear enough to start to grow crops and rebuild.
Humans, as well as many other creatures, will become more radiation hardened over generations, and given the bountiful supplies of tech and resources left over from our civilization, will be able to quickly repopulate the world.
https://www".dailymotion.com/video/x4e30ki
The punchline is at 29minutes
I said civilization would collapse in a dire nuclear war scenario, which is not the same thing. Advanced civilization itself may actually be quite fragile - we see glimmers of that fragility now and again when government / order / utilities break down for prolonged periods.