There are two major ways in which nuclear weapons and their parts have been built. Smaller pieces of support systems were sometimes produced by regular industry in their own factories, under contract. Most critical components however were developed at what are known as GOCO (government owned, contractor operated) facilities, like the Pantex plant in Texas, or Y-12 in Tennessee. These are government facilities, but they are operated by private contractors. (A complicated exception to this is that the weapons laboratories themselves, until the early 21st century, were operated by a public university, the University of California.)

What this means varies over time. One of the earliest contractors, DuPont, did their war work for essentially no profit (because they were worried about being considered war-profiteers). Over time these became more profit-driven, with all of the sorts of issues (poor oversight, etc.) associated with government-private industry collaborations.

As for which companies, it is a very, very long list. Most of the big contractors were big industrial companies well before they got involved in the nuclear business. Honeywell; General Electric; Westinghouse; Bechtel; Monsanto; Bendix; Rockwell International; DuPont... at various times, various contractors. Many of them are still around, some of them have been folded into other conglomerations (Rockwell International split up into other companies; Bendix was acquired by another big company; etc.). It is tricky to keep track of them over the sprawling nuclear complex, and many of the specific jobs (e.g. security) are sub-contracted out. (The feds at these sites are known as "WeBes" because "we be here when you come, we be here when you leave" — there is a lot of swapping around contracts.) For a contemporary approach to identify money that ultimately contributes to the nuclear complex, a group of anti-nuclear activists have made an app that tries to indicate which funds are invested into companies that develop nuclear weapons.

The production of the arms has not been under military control since the passing of the Atomic Energy Act of 1946, which explicitly requires it to be under civilian control. There is interesting history here as to why it was done this way. Even the Manhattan Project, though, while definitely under military control, involved heavy use of industrial contractors to build and operate the facilities in question.

I don't know if there are any figure for which companies made the most. The accounting regarding Cold War weaponry was not as complete as one might like (the government kept no ongoing ledger for the costs of nuclear weapons, for example), and has often been classified (a practice that has not gone away). The best discussion of these issues is in Stephen Schwartz, ed., Atomic Audit (1998).