An ultimatum — or threat — is only as good as your ability and desire to go through with it. This is known as "credibility" in the literature on deterrence. Would the US have been able to make a credible threat or demand to the Soviet Union that the Soviet Union would have heeded? The answer is: yes, but it was a very limited one.

The US nuclear arsenal, until the 1950s, was very small, especially relative to the size of the targets in the USSR. These early bombs were also not easy to deploy — it was a bomber-based arsenal, and to use bombers of that vintage you need bases relatively near to your targets. The US did not have such bases for the most part until after the Soviets had their own nuclear weapons (here is a map I made of US foreign nuclear weapons deployments from 1945 until 1958). Now some of this could have been done differently if there had been a will to do so — notably the number of bases.

But some of these conditions are related to material infrastructure: one cannot spin up thousands of bombs instantly, it requires deep infrastructure investments into the bomb-producing facilities. The WWII era facilities were set up to produce about 3.5 bombs per month, as a point of reference. Changes in bomb design could expand that number to maybe 6 bombs per month. In practice, the facilities weren't capable of operating at full capacity for a lot of that time anyway (the Hanford reactors suffered production problems that needed to be worked out, part of the consequence of them being the world's first industrial nuclear reactors). Starting from 1947 onward, the US Atomic Energy Commission began major investments in expanding its production capabilities, which came to fruition in the increased capacities in the 1950s that produced the swollen arsenal of the Eisenhower era.

Separate from this is the question of what the US goals and priorities were. The nation was weary of years of war. The atomic bomb — at least not in its state at that moment — was not a weapon that could take out the Soviet Union overnight. They simply lacked the capability. A war with the USSR would be like World War II but occasionally the use of Nagasaki-style bombs. There would be no surprise to that; the Soviets knew about the atomic bomb, unlike the Japanese, and would take efforts to destroy such planes before they reached their targets, and would also disperse their industry and governance in ways that would make them hard to target. The Red Army was massive, and already had a good chunk of Europe under its control. They would undoubtedly attempt to take the rest of Western Europe, to deny the United States a launching point for attacks of any sort. Perhaps the US would win in such a situation, but at what cost? What ruin would come to the already-ruined Europe? Would the US population be willing to pay the cost in lives and money for such a thing, without an outright provocation? And what of the moral aspect — would a significant majority of American population support a preemptive and (in the eyes of many) unprovoked war of this sort?

The answer is plainly "no" to all of these questions, and certainly there was no appetite for it by Truman. What was desired was a return to peacetime and newfound economic prosperity. No, they didn't want the Soviets to take over Europe or anything else. Nor did they want them to have nuclear weapons. But preemptive war was off the table, at least among anyone representing a kind of mainstream politics.

So what options were there? One was what would be called "containment," which was to say, "we aren't going to wage active war on you, but we will do everything in our power to prevent your expansion, and under some circumstances we would consider war against you, but only if you cross some very specific line." So if the Soviets had tried to take over West Berlin, that might have triggered such a war. As it was, the Soviets toyed with that line, testing what they could get away with — the Berlin blockade, for example. But the Soviets didn't want a war, either. Hence the "Cold War" — a situation in which the two superpowers wanted to avoid head-on fighting, but were nonetheless antagonistic towards each other. This does not mean there was peace — the Korean War and other proxy wars killed millions — but it means they avoided fighting directly.

The other option was to try and use diplomacy to achieve the US goals. Notably in the area of nuclear weapons the US did try, through the United Nations, to see if there was a way to ban the development of nuclear weapons by any country. This did not go too far, but an effort was made.

Even after the Soviets tested a bomb in 1949, the US still maintained a massive advantage against the Soviets. Until the 1960s the Soviets had essentially no credible way to threaten the continental United States with nuclear weapons; they could only threaten US allies in Europe. Even this was enough to make the gains of any war with the Soviets not worth the costs.

A brief note on China: China did not become an opponent of the United States until 1949, with the Communist revolution there. The US did threaten China with nuclear weapons in the late 1950s with regards to the Taiwan Strait Crisis. But the very large conventional forces in China similarly made the costs seem very high, even without assuming that the Soviets might take it as a line too far of their own at that point. The threats did, however, motivate the Chinese to start their own nuclear weapons program.

In general, people today overestimate how useful the atomic bomb was as a military "tool" in the 1940s, and also misunderstand how the calculus of risks and benefits worked. They also underestimate how uninterested the people of the US were in the post-WWII period of jumping into another war (the Korean War was extremely unpopular, as an example, and it was much more limited than any direct war with the Soviets or Chinese would have been).

A very interesting book that explores many angles to exactly this question is George H. Quester, Nuclear Monopoly (2000).