There are two separate technical issues you are talking about here. One is the number of atomic bombs capable of being assembled rapidly. The other is their ability to be quickly deployed to targets.

In terms of bombs, the DOE has released figures indicating that by 1948 they had nuclear and non-nuclear components for some 50 atomic bombs. But this is not the same thing as them being assembled and ready to use. The atomic bombs of 1948 were exactly the same as the Fat Man bombs dropped on Nagasaki, full of fiddly parts that require experts to put together and are easy to get wrong.

And they were probably not assembled in any great numbers. The bombs of that vintage could not be assembled and stored for long periods of time; their initiators needed to be replaced fairly frequently and they were hard to take apart and put back together.

The bombs of this time, as an aside, were not even in the possession of the military! This was still in the period where the civilian Atomic Energy Commission retained full physical control over the weapons. This only started to change, slowly, starting in 1948, when Truman authorized the military to have control of some non-nuclear assemblies. They did not get nuclear components until 1950.

Starting in 1949, a new bomb, the MK-5, came onto the scene. This was a much improved version of the Fat Man bomb — it was much more "G.I. proof" as they put it, in that it could be readied to use relatively quickly. It also was designed to allow easy insertion of the nuclear pit, which meant the temporary improvement of the custody situation (the AEC could just keep the pits). But in the 1950s, the use of sealed-pit weapons, and machinations by the military, resulted in nearly all US nuclear arms being in their possession.

As for carrying out the attack, the US did have long-range bombers (B-29s until 1949, then B-36s) and close bases to the USSR. However they did not deploy weapons to overseas bases until around 1950. They eventually built quite a ring around the USSR but that came mostly in the 1950s.

So, anyway, you really have two questions here. First question: Would the USA have been able to carry out a massive nuclear attack in, say, 1948 or 1949? Not instantaneously, no. Atomic attacks were still in the "takes several weeks to carry out" phase. They were logistically tricky. Prior to 1950 or so I think the kind of war you'd be looking at would be a mostly conventional war punctuated by occasional atomic bombings, not the "over in a matter of hours" sort of nuclear war that became possible starting in the later 1950s.

Second question: Was Truman bluffing or a dope? It is very hard to read Truman's mind retrospectively. His understanding of things nuclear was often tenuous at best. He did not often write his own speeches on this matter, and later came to contradict much of what he said in such speeches on atomic matters. So I really can't say. If I were to guess I would say he probably didn't know much about American capabilities and probably didn't care very much about the nitty gritty logistical aspects.

On the Custody Dispute, I have written about it at some length here, with a link to a lengthy, redacted report on it. Generally speaking, the rapid strike nuclear capability that we have come to associate with the US during the Cold War did not become an operational reality until the early 1950s.

If Bulletin of Atomic Scientists is credible enough to post here, then yes. According to this report, it became a very realistic threat to make as the weapon count exceeded a dozen, reaching 50 by 1948. Now, regarding the reliability of those weapons, or probably more pertinently, the bombing practices which would have been employed in delivering them, that's another story.