The only reason the UN was even minimally effective was because it provided a venue for the nuclear powers -- who almost exclusively make up the Security Council -- to hash out problems diplomatically. Without the constant threat of nuclear war to bring those parties to the table, literally and figuratively, there's no reason to think that the UN would have been any more effective than the League of Nations.
And the League, as you'll recall, was also set up in the aftermath of a staggeringly destructive war, by a great number of very committed people, and it couldn't and didn't do the job. In other words, the memory of the utter hell that was the Great War wasn't enough, even among people who had lived through it, to keep the peace through diplomatic methods alone. However, the threat of total global annihilation has kept things in check for more than three generations now.
Given how close the Cold War came to going 'hot' even with nuclear weapons making it into a no-win situation, it's laughable to suggest that we wouldn't have gone there in their absence -- when either side could have talked themselves into believing that they could have obtained a real advantage by fighting.
The horrors of conventional war have never been enough to keep people from deluding themselves into thinking that it can be won (because, bluntly, it can be); nuclear war is unique in that it is quite obvious that there can be no winner, and it is to everyone's advantage to avoid, all the time.