NATO becomes the EU's military, that's what. And we end up with World War IV. (GWOT being World War III, which hasn't really ended.)
Actually, the Cold War was WWIII, fought in slow-motion, without the principals directly engaging (except for a few dogfights over Korea) thanks to nuclear deterrence. Lack of strategic vision made people think it was lots of little wars, each of which was actually just a campaign: the Greek civil war, the Berlin airlift, Korea, the Cuban Revolution, the Cuban Missile Crisis, Vietnam, the Contra war in Nicaragua, the civil war in Angola, the liberation of Grenada, the Russo-Afghan war,.. were all just campaigns in WWIII.
I'm not sure the GWOT actually counts: only one major power involved, even if it's got engagements on most continents.
(Of course, by the standards by which WWI and WWII were called 'world wars': engagements on most continents, all major powers involved, the Seven Years War (French and Indian War in these parts) and the Napoleonic Wars (War of 1812 in these parts, and yes, we were on Napoleon's side geostrategically) should have been called "world wars'.)