The US nuclear complex at the height of the Cold War was quite large. The facilities could be (and were) divided into two categories: 1. specialized government-owned, contractor-operated (GOCO) production facilities that produced only nuclear weapons, and 2. contractors that produced components as part of their regular work.

This model goes back to the Manhattan Project. There are some kinds of things that because of their nature or their security cannot be meaningful produced at regular industrial sites. An obvious example of this is plutonium, which requires massive nuclear reactors and nasty chemical processes to extract the plutonium from spent nuclear fuel. This was produced by the Hanford Site in Washington and the Savannah River Site in South Carolina. Both of these, like all production facilities, were GOCO: they were federal sites but always operated by contractors (Hanford was initially operated by DuPont, then General Electric, later Bechtel, etc.).

Even the nuclear weapons labs (the places where scientists developed weapons designs and did research) in the United States are run on such a system; Los Alamos and later Livermore were GOCO, run for years by the University of California (only fairly recently did it switch to a consortium of private and university contractors).

Wikipedia has an OK list and map of the major Cold War sites. There were of course many more minor subcontractors (for the Manhattan Project, I have counted several hundred subcontractors — these ranged from places that might just be providing standard laboratory equipment or even office furniture, to places that might produce specialized feed materials like big, refined pieces of bismuth or even uranium metal).

If I were trying to characterize the "flow" of these sites, it looks something like this:

• You have sites that produce and refine raw materials, like uranium ore. (Private companies, e.g. U.S. Vanadium Corporation.)

• You have sites that take the raw materials and turn them into something you can use in the processes of fissile material production (e.g. turning uranium ore into uranium metal, or into uranium hexafluoride). (Initially private companies, later GOCO facilities; e.g. Fernald Plant.)

• You have the sites that produce fissile material. (GOCO facilities; e.g. Hanford or Oak Ridge.)

• You have the sites that research and test weapons designs. (GOCO facilities; e.g. Los Alamos, Livermore, and Sandia National Labs.)

• You have sites that fabricate the non-nuclear parts of the weapons, like the high explosives, electronics, or internal metallic components. (Initially private, later GOCO; e.g. Kansas City Plant).

• You have sites that fabricate fissile materials into the required shapes, and do other dangerous and highly-sensitive work. (GOCO facilities, e.g. Rocky Flats.)

• You have sites that assemble the nuclear and non-nuclear components into weapons, or disassemble retired weapons. (GOCO facilities; e.g. Pantex.)

When the weapon is produced, they ship it to wherever it is destined to be. This could range from the Nevada Test Site, if they are planning to detonate it, or to storage bunkers, or to the sites where the delivery vehicles are kept. Typically this has been by means of a truck convoy. If you do a Google Image search for "nuclear weapon truck convoy" you can see what these look like. They are pretty big trucks followed by other trucks and cars and probably have a lot of guys with guns in the back. Whether this is adequate security or not depends on what you think is going to be the security threat. In the 1960s there were lots of fears that a sufficiently large and organized force of terrorists or something could overwhelm such a truck, because there would be long stretches where the truck would be out of communication range with any other backup forces. Today that seems less likely. In any case, the weapons have been designed for awhile now so that they cannot be used or tampered with without special equipment (PALs).

There is an entirely separate complex for developing the delivery vehicles (bombers, rockets, etc.). Typically these are produced by private industry, e.g. Boeing and Lockheed and other big big big aerospace contractors who have been working (in various forms) for the military for a long time. (The delivery vehicle research, development, and production cost considerably more than the production of the warheads.)

Delivery vehicles are kept separate from the warheads until they are "mated" (technical term!). This is done at the delivery sites (e.g. missile bases) by the military.

The US complex is pretty complicated in terms of organizations, because there is a weird GOCO system to begin with, but also because there is a legal split between the civilian (Atomic Energy Commission/Department of Energy) and military (Department of Defense) with regards to production and "custody" of the weapons. In the early days this meant that the warheads were technically not given to the military unless war was thought to be eminent; they were kept on base in separate, civilian-run facilities who had instructions to shoot military men who tried to illegally take custody of the weapons. By the late 1950s technical changes in the weapons and a fear of this getting tangled in a rapid response meant that the military was given possession over almost all of the weapons. This is known as the "custody" dispute.

I know the Soviet system in less detail, but they had a similar demarcation of roles in their facilities, and many of their facilities were "copies" in a general sense of the US way of doing things. The Soviets did not have GOCO of course, or as firm delineations between civilian and military, but they were not as different as you might think. The Soviet equivalent of the AEC/DOE was the Ministry of Medium Machine Building (known by its acronym Sredmash), a purposefully cryptically-named part of the government concerned with nuclear weapons development. Their equivalent of the DOD was the Ministry of Defense, and notably the Strategic Rocket Forces created in 1959. Their sites were similarly spread across the country, operated as "Design Bureaus" that competed with each other, similar to how some of the big aerospace companies and national labs competed with each other for contracts. They had similar "pipelines" of materials and components and actual weapons, though I do not know the details of that.

The best breakdown of the US complex and how it evolved over the years that I know of can be found in Schwartz, ed., Atomic Audit: The Costs and Consequences of U.S. Nuclear Weapons Since 1940 (Brookings, 1998). It's an attempt to really catalogue exactly how much money was spent on the complex and where it was spent, and that involves going over the whole complex. For the Soviet side of things, Podvig's Russian Strategic Nuclear Forces has a lot of information.