The Atomic Energy Act of 1946, which was passed in early 1946 and went into effect in 1947, create a regime for domestic control of nuclear weapons that was explicitly monopolistic. It could not be any clearer on this point. Aside from fact that it prohibits private ownership of fissile material (bomb fuel), fissile material production facilities, and even declares all information relating to fissile materials or weapons designs automatically classified (even if you independently discover it), it very clearly states in Section 6(b):

It shall be unlawful for any person to manufacture, produce, transfer, or acquire any equipment or device utilizing fissionable material or atomic energy as a military weapon, except as may be authorized by the Commission.

The regime of the Atomic Energy Act was one of total monopoly, giving the US Atomic Energy Commission control over pretty much every aspect of the fuel cycle and weapons production. It allowed for some private participation in this process, but only through a series of licenses that were overseen by the government. In practice you could not even do research on these things without special clearances and licenses.

There were substantive revisions to the Atomic Energy Act in 1954 which loosened some of these restrictions, with the aim of allowing the private sector to do research into aspects of nuclear technology that had peaceful implications (which could include dual-use technologies, like nuclear reactors and uranium enrichment). But these were again through a series of licenses and clearances — they just made some of these easier to obtain.

In terms of whether they feared it could be done: definitely, yes. This was explicitly on the mind of Congressmen as they contemplated making this legislation, and can be seen in aspects of their public hearings in late 1945 and early 1946. The issue where this came up in particular was their obsession with private patents of nuclear weapons, which they ultimately outlawed entirely. (Nuclear weapons are one of the few categories of technology in which patenting is disallowed in the United States. It is the only such category where Congress made that determination, to my knowledge. As such this fact comes up whenever there are questions about patentability of new technological concepts, like biological life.) If you are interested in how the question of atomic patents got on their mind (and what they made of it), see my article from a few years back: ["Patenting the Bomb: Nuclear Weapons, Intellectual Property, and Technological Control"](http://alexwellerstein.com/publications/wellerstein_patentingthebomb(isis).pdf).

Taking off my history hat and putting on my policy hat: as for whether a private citizen could do it: it depends on what you imagine they are "doing." Are they buying black market highly-enriched uranium or plutonium and then making a weapon out of it? That would be entirely within the means of a wealthy individual, if they didn't mind being caught. (This would be considered a form of a nuclear terrorism.)

If you mean, "could they duplicate a fuel cycle that led to them acquiring fissile material and then making a weapon from it?" The answer is: this would be pretty hard to do, at least without being noticed. You are talking about a substantial industrial enterprise here. There are certainly private companies that have the resources to do this — big engineering firms like Bechtel (which has an annual revenue of approx. 10X that of SpaceX, as a point of comparison). But it'd be pretty obvious what they were doing and it's hard to imagine a total lack of whistleblowers.

The most plausible in-between situation would be companies that have access to fissile material production through their regular operation diverting or "losing" material. This is entirely plausible and may have already happened at least once (see the NUMEC affair). More broadly this is considered a problem of "insider threats." Could it be done without detection? For some plants, certainly: the amount of "lost" material could be within the threshold of detection (known as "Material Unaccounted For") but still be a significant amount of material (from a weapons perspective) at high-volume plants.