Well after WWIII most of the politicians and political systems that kept us separate were destroyed, so any corrupting or nationalistic elements from the old order would've gone up with them. Plus I'd say that enough people were killed off that humanity had to become more interdependent on each other considering the state Earth was in to survive especially with the greater universe staring them down.
So all that together, plus 100 or so years between FC and ENT would be enough to get us on the track to a more United Earth.
The problem is that nationalism still exists even in 2151. Remember, the British Royal Navy still exists in some form in the 22nd century. That implies that United Earth still is not quite developed. (this is one of the problems that I had with Enterprise as a whole, because I don't think they ever took the time to genuinely consider what 22nd century Earth was really like).
I suppose one could argue that United Earth was an out growth of WWIII (Just as the League of Nations was an outgrowth of WWI and The United Nations was forged in the fires of WWII). Ultimately, the answer to this question I suspect lies in the nature of WWIII. Lilly's comment about "factions" in 2063 indicates that WWIII Earth had become very divided. The DS9 Episode "Past Tense" paints a picture of earth in 2029 that was gradually slipping into chaos. The US constitution clearly was not fuctioning properly and there were reports of Neo-Trotskite riots/revolutions breaking out all across Europe. Its likely that the nation state system started breaking down, which is why the New United Nations was not in a position to manage the situation. If we throw in the possibility that there was also a religious aspect to some of the conflict, and the left over paranoia from the Eugenics War....WWIII makes alot of what we seen in 23rd/24th centruy Earth seem more plausible. A world wide holocaust caused by rampant tribalism/nationalism and religious fervor might explain why these things vanished or diminished in the aftermath of the war. They simply burned themselves out.
Remember, after WWII, amongst the people themselves (if not strictly speaking the governments) there was a sense that things had gone too far. The Brutality of the war led many people to want an institution that could make such wars impossible. The United Nations was supposed to be that body. It was initially concieved of as an embryonic world government. It had legally binding decisions and even the ability to wage war. The European Coal and Steel community was born for much the same reason. The UN, as it turned out, failed to be that government. Once the dust of war cleared and civilization picked itself back up, the old nationalism reemerged though outside forced (i.e. the US and USSR) worked to keep nationalism in check in Europe (thus creating the space for the EU to eventually emerge). However, the only reason the world recoved as fast as it did is because, for all the fighting that it did, the US and the rest of the nations of the Americas, were left untouched by the war. This left an industrial, educational and economic base intact to facilitate recovery. In the aftermath of a global thermonuclear war (which was itself preceded by a devastating war...and then followd up by a campaign of genocide by Colonel Green.....its not hard to imagine that an exhausted and impoverished population had had enough of war, fighting, religion, nationalism etc. They just wanted peace and the return to some sense of stability. Once the Vulcans arrived and brought the material support necessary to facilitate recovery (as well as the schocking revellation that humans are not alone and that there vast numbers of civilizations...both friendly and hostile "out there") the human race found a reason to band together. No doubt the Vulcans, with their impecable logic, also encouraged a woundeed but healing humanity that unity (and logic) was probably the best way to keep such a catastrophe from happening again. A truly traumatised world would finally be willing to listen and heed that advice.
Much as WWII had a deep emotional effect on Europeans, and made them far more adverse to war and open to cooperation, it seems likely that WWIII had the same effect on the rest of the world.
