At the most, a few thousand people may be able to escape (temporarily) in space shuttles and space stations. But their long term survival is in serious doubt. There will be a scarcity of food and water. The survivors will cannibalize each other in the end.
What is the cost of manufacturing a space shuttle? $100 billion perhaps? We may be able to manufacture them at $10 billion a piece, if mass production is done. And the total net wealth of all the humans right now is around $300 trillion. Just calculate how many shuttles can be created.
Yeah that is as far from reality as we can get. If Ww3 commences, there's pretty much no hope for us and we're all eff'd up. No escape especially for regular people who comprises most of the population. I think it now rests on the world leaders to know their responsibilities and to know that a third world war can very well mean extinction for mankind.