Remember when the US and USSR were both using German rocket scientists? The local brains may be great brains, but they aren't necessarily the best brains.
So three countries tried and only one got it right. But in the end everybody benefited, even without the war the others would be inspired by their work or simply copy it.
A single world government would most likely run a single project with one in three probability of success, judging by the real success rate.
And the Germans weren't the best brains, they simply had a lot of luck that they project were supported and financed. The Americans and the Russians scientists were similarly talented, but their governments saw no need for such weapons earlier, during the war.
And I don't follow how working independently made the "free world" a game winner.
They didn't work entirely independently, they pulled together resources when needed - for example in defense.
The communist bloc pulled resources for most trivial reasons, creating single mandatory designs for all their members, so when mistakes were made, and they happened frequently all members were equally affected by them.
Like the infamous ES EVM, a single computer system for all (a stolen IBM design), that was forced on everybody despite the fact local, frequently superior systems were available.