WWIII will not be a child of those Wars.
The name of the war is an interesting topic. Obviously, WWI (or First World War) wasn't used at the time that war was being fought; it was called by most people the Great War. In the book Europe's Last Summer, the author makes the case that WWI and II aren't properly even separate wars; WWII was just the conclusion of the Great War, which had gone cold in the 20 years since the Armistice, and he therefore detests the name WWII, since it makes one think that the two can be considered separately. I would consider them both the Great War, but make the distinction of The Kaiser's War and Hitler's war, since it was they who instigated.
The way I see that it could go down, is that some regional conflict (Iran, Syria, etc.) starts to spread, everyone becomes worried that the next geopolitical structure might be very different, and they dive in just to see that their interests are represented, -if the UN can't prevent it them from doing so, that is. It's so popular to slam the UN as being ineffective. But I see the tangle created by the UN as a powerful force of global stability. As long as China and Russia are in, everything eventually becomes too complicated.