The jump from the discovery of fission, the phenomena, to a usable weapon was indeed a large one, albeit one that could be accomplished in a short amount of time once the commitment to do it at any cost was made.

The major difficulty lies in the fact that to make a fission weapon you need very specialized materials to do so: either uranium that is very enriched in the rare uranium-235 isotope, or you have to create plutonium-239, which is made in a nuclear reactor. (There are a few other elements you can use as well, but the difficulties only get harder beyond these two methods.) They discovered that regular old uranium as mined from the ground would not work for a weapon in 1939 (it can work in a reactor, with a clever setup), and so a lot of the discussions from that point onward were about how difficult it would be to separate out the uranium-235 isotope, and how much of it you would actually need. (The idea of plutonium came a little afterwards, and confirmation of its existence and properties came in 1940.)

Uranium isotopes all behave the same, chemically. They're all uranium. They differ only in a tiny difference in mass. Most (+99%) uranium found in nature is in the form of uranium-238, which not only does not sustain a fission chain reaction, but actively inhibits such a reaction. So to use that as a fuel for a bomb, you have to separate out a lot of the <1% uranium-235 and in very high purities (the uranium fuel needs to be >80% uranium-235 to be a reliable weapon). Separating isotopes had been done prior to attempting it with uranium but because the difference between the two isotopes is very small (about 2% of mass) it is very hard to do for uranium and to do on a large scale. The factories involved in doing this consumed some 60% of the resources of the Manhattan Project and require the labor of tens of thousands of people to construct and operate. This is an industrial-scientific problem — there were scientific theories to how to do this separation, but scaling them up was very difficult. One of those enrichment factories, the K-25 gaseous diffusion plant, was the largest factory under one roof in the entire world at the time.

To make plutonium-239, you need an operating, industrial-sized nuclear reactor. These did not exist prior to World War II, and the first even research-scale nuclear reactor did not go online until late 1942. They immediately took the lessons learned from the small-scale reactor and used them to start building three reactors that were many times larger, as well as all of the chemical facilities necessary to separate plutonium-239 from the spent fuel of the reactor. Again, this was a gargantuan effort, and also involved totally new engineering work — they were literally going from laboratory-scale to industrial-scale in one jump, on a totally new technology. The cost of building and operating these reactors took up some 20% of the Manhattan Project budget.

The rest of the work on the project was largely about aiding these methods, and for working out details about the bomb design itself. While you can use uranium-235 in a very crude weapon (smash enough of it together at once and it'll explode, the essence of the "gun-type" design), plutonium-239, they discovered, requires a much more sophisticated approach. Developing the implosion design, in which a solid ball of metallic plutonium is instantly compressed to over twice its original density using extremely symmetric high explosives, required the labor of thousands of scientists.

And of course working out the specific applications of the theory — exactly what will happen if you detonate a bomb with X characteristics — is much harder than knowing the basic theory. This requires experimental data (like very refined fission, scattering, and absorption cross-sections of the fuel isotopes), and very careful calculation. The number of "unknowns" in the 1940s were huge. They didn't realize that plutonium-239 could not be used in a gun-type design until the summer of 1944 — a pretty big "unknown"! — and so only had a few months to develop the implosion design from scratch. Plutonium is one of the most chemically- and metallurgically-complicated metals that we know of, and they didn't get their first macroscopic samples of it until well into the project.

There were other aspects to the project than the above (such as working out the health safety aspects of these new materials and processes, and developing some of the more esoteric aspects of the bomb, like the polonium used in the neutron initiators), but they constitute the major hurdles towards making a nuclear weapon, then and now. It's not about theory. The theory is relatively straightforward. It's about turning that theory into a reality, which in the case of nuclear weapons means making the fuel and the weapon design, and these were just genuinely difficult scientific, engineering, and industrial problems. Over the years some of these have gotten easier (reactors are not exotic anymore; centrifuges simplify enrichment issues dramatically; modern computation capability makes some of the hard design problems much easier to simulate) but any aspiring state would go through similar considerations today in thinking about how to become a nuclear power.