In nuclear strategy, there is a concept called "credibility" that means more or less what one thinks it does: if you make a threat, it has to be credible that you'll follow up on it if it going to affect whomever you are making it to. So, as just an example, Putin has claimed that if the US and Europe continue to issue sanctions against Russia, he might nuke them. Is that viewed as a credible threat? Clearly not: the US and Europe have kept their sanctions in place, because they don't believe that Putin would actually risk full-scale nuclear war over economic sanctions.

So let's look at whether the US could have used nuclear war as a possible threat to stop proliferation. In the case the USSR, it's not clear they could. First of all, during the period in which the US had a nuclear monopoly (1945-1949), the US nuclear arsenal was still quite small and hard to deliver. It was not being actively deployed to foreign bases, and the US would need those if its bombers were to reach Soviet targets. The arsenal was also considerably smaller than what the US thought was necessary for a full, all-out, nuclear-only attack that would disable the Soviet capability for war (several hundred weapons). So it wouldn't be some kind of one-shot thing, it would be a prolonged war that would possibly lead to incredible destruction in Western Europe as well. The US population didn't want that — they wanted to demobilize — and their European allies didn't want that. And that kind of threat would justifiably disturb even allies about the US intentions regarding being a world, nuclear policeman. It would not exactly be in the spirit of the new United Nations. So you can see why all of this would have been a very non-credible threat.

What about other nations? Again, is the US likely to — do what, exactly? Use weapons against cities if they don't get their way? Destroy nations? Invade and take over? It's a tall order. When you start to look at it concretely, it seems rather absurd that the US would even want to approach things this way, much less that anyone would ever have responded the way they wanted. Would they have credibly threatened the UK or France with such destruction? How would other European powers have felt about that? The same European powers, incidentally, that the US needed the favor of to place its nuclear bombers near Soviet targets.

The US actually did threaten China with nuclear attack, in 1958. Not over proliferation, but over their actions in Taiwan. China did back down... but it also immediately started its own nuclear program, in secret. So you can see how these kinds of threats can produce results that are not what the threatener might want, and how the short-term victory can lead to a long-term problem.

What the US did do after WWII was this: it attempted to get international support for a treaty that would ban all nuclear weapons, including its own. This saw little traction because the Soviets were not that interested, because they didn't trust the US and felt that the only way to guarantee their own security was to have their own bombs. The US was not initially that disturbed by the UK getting the bomb, because they were an ally. They would have rather France didn't, but it seemed like something they had little power to control. They had no real way to influence the Chinese decision, unless they had wanted to wage war with China, perhaps, and that was something the US had no interest in doing, especially since for part of that time the Chinese and Soviets were closely aligned.

There were places where the US tried to exert influence to prevent proliferation, sometimes with positive results, sometimes not. Israel was a case where, in the 1960s, the US tried to discourage them from going nuclear, but failed (and eventually the US just agreed to live with it). They had better luck discouraging Taiwan and South Korea from going nuclear, threatening to withdraw their security guarantees and military support if they did. This is sort of instructive: you probably can't bully an enemy into disarming (it'll produce fear, not security), but you can convince an ally that it might not be in their interest to cross you too much.

The US had other approaches with European allies, including bilateral assistance treaties that allowed them to both guide the direction of their nuclear research programs (towards peaceful ends) and encourage a state of dependence on the US (for fuel, etc.), with non-proliferation being one of the goals of those efforts.

The Nuclear Non-Proliferation Treaty (NPT) was developed as a diplomatic means of trying to enforce nonproliferation, and it is notable in that it is not really based on threats so much as promises of assistance and mutual agreement. It's not perfect but it arguably has a better track record than a threat-based method.

Anyway, there is more that one can say about this, but hopefully you get the drift. One might think that having the bomb gave the US a whole lot of options it did not have before, but in practice, it didn't do as much as they had hoped. Eventually the US policymakers and strategists came to the conclusion that nuclear weapons really only did one thing: they kept other nations from attacking you overtly, especially with nukes of their own. Which is something, to be sure. But it is not a blank check for any kind of power, unless you're actually willing to use them, and even then, to do so would have consequences that probably would not be worth whatever gains one was trying to get.