Once you figure out how to make them, nuclear warheads can be factory produced and assembled. The US during the late 1940s and 1950s built up the infrastructure, a nation-spanning industry, to produce nuclear warheads the same way that Ford produced automobiles: at scale, and routinely.

The biggest bottleneck was not the actual assembly or even fabrication of parts, though those did have some dedicated facilities like the Pantex Plant, where most US nukes were assembled. The hardest part was creating enough enriched uranium and plutonium for the warheads. That is the expensive and difficult component, and it requires a huge amount of raw input in the form of uranium oxides. The US worked incredibly hard to secure both international and domestic stockpiles of uranium, going as far as creating an artificial "Uranium Boom" in the mid-1950s to encourage uranium prospecting, and by the 1970s had more uranium than it really knew what to do with. They also built several gigantic facilities for enriching uranium, and several for running the industrial-sized nuclear reactors to make plutonium.

In the 1950s, during those years of the gigantic stockpile increase under Eisenhower, the US Atomic Energy Commission had many very large dedicated facilities devoted to producing the various parts for the bombs, including but not limited to:

• Los Alamos, Livermore, and Sandia National Laboratories, who designed, tested, validated, etc., the warheads

• The Nevada Test Site and the Pacific Proving Grounds, where they were actually tested

• Fernald, which turned uranium into reactor fuel

• Pantex, which assembled weapons and fabricated explosives

• Oak Ridge, Porthmouth, and Paducah, which enriched uranium

• Hanford and Savannah River, which produced plutonium and tritium in nuclear reactors

• Rocky Flats, which fabricated plutonium pits and uranium bomb parts

• Mound, which produced initiators, detonators, timers, and other bomb parts

• Kansas City Plant, which produced electrical parts

And so on and so on. It was a huge complex with thousands of people working in it. This Google Map of the US Cold War Nuclear Complex gives some sense of the scope of it.

I've said it already, but I want to reemphasize: this was the creation of an entire multi-billion dollar industry dedicated to one major output, owned and funded by the US government. That is how you end up in a situation where you can produce thousands of nuclear warheads per year. Right now, we couldn't do that if we wanted to, because at the end of the Cold War most of that infrastructure was mothballed.

In terms of the cost relative to the defense budget, it's very hard to break out nuclear weapons from the rest of it, because that isn't how the records were kept (and the weapons are more than just the warheads), but the total cost of the US nuclear complex during the Cold War is estimated to have been on the order of $5 trillion dollars in late 1990s currency. Which is huge. However, as a share of total US defense spending, it was still pretty small. Because as expensive as that is, it's not as expensive as something like the Vietnam War. Which is more a commentary on how much money the US spends on defense (currently around $700 billion/year) than it is on the cheapness of nuclear weapons.

A very nice overview of the evolution of the Cold War nuclear complex is Charles Loeber, Building the Bombs: A History of the Nuclear Weapons Complex (Sandia National Laboratories, 2002). On the cost, Stephen Schwartz, ed., Atomic audit: The costs and consequences of U.S. nuclear weapons since 1940 (Brookings Institution Press, 1998), is still the best reference.