There are several elements to it.
First, you need to decide which version of a nuclear bomb you want to build. We’ll leave fusion bombs off the list for now, because you have to master at least one of the other types of nuclear weapons to build a fusion bomb (fusion bombs require a fission bomb to create the conditions for fusion).
The simplest type of nuclear warhead is called a gun-type bomb. These are actually rather simple to construct: You just need two sub-critical masses of Uranium-235, enriched to very high purity. Load them into a bomb container, separated by some distance. Fire one mass at the other at high velocity, and provided you have a large enough mass and velocity, it will start a chain reaction and give you a nuclear weapon. This is the type of warhead that was dropped on Hiroshima.
The problem with a gun-type warhead is the two masses of U-235. The weapon itself is simple, but refining your sub-critical masses is very hard. You need quite a large amount of it, and regular uranium is mostly Uranium-238, which can’t be used to make a bomb. And since both isotopes of uranium are chemically identical, the only way to refine them is to take advantage of the small difference in mass to separate U-235 from U-238. It’s easier to do these days thanks to high-strength centrifuges, but it’s still not something you can just do on a whim.
The other type of nuclear bomb is called an implosion-type bomb. This type of bomb uses a plutonium mass, which is carefully built into a perfect sphere whose density is just below critical density. It’s then surrounded by explosive lenses, which are carefully timed to detonate at exactly the same time. The resulting explosion then compresses the plutonium enough to cause it to go critical, giving you a nuclear explosion. This is what was detonated in the Trinity test, and also what was dropped on Nagasaki.
Getting the plutonium is much easier than U-235, because plutonium is one of the natural decay products from a fission reactor, and the isotope that’s created is suitable for a nuclear weapon. Separating it out from the nuclear waste is simple since it’s chemically distinct from the other waste products. However, building a proper explosive lens is much more difficult, and even the slightest misalignment will give you a fizzle instead of a proper nuclear bomb.
In short: No matter what, you need to do some sophisticated engineering of extreme precision in order to build a nuclear bomb. The countries that have managed it have all either developed or were given a number of tricks and refinements that either lowered the amount of fissile material requires or made it easier to construct the lensing effect necessary for the fission reaction. Most countries were given the secrets or stole them from other countries in order to manufacture their bombs; developing them from the ground up requires engineering skills that are difficult to find.
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