As noted, the idea of nuclear fission was well understood (at least theoretically) since the 1930's, and indeed the idea of an "atomic bomb" had been around even earlier (H.G. Wells postulated what we would likely call a "dirty bomb" today in "The World Set Free").
There were actually 3 major secrets to the atomic bomb, but even then they were penetrable if you were willing to expend the resources necessary to do the experimental work.
Secret number 1: How do you extract and purify the fissile material for the bomb? This is a question which delves more into the realm of heavy engineering, and since the answer wasn't clear, the "Manhattan Project" went to the great expense of trying out every method to discover which one worked best. If the American and British nuclear programs were not infiltrated by spies, this would likely have set the USSR and everyone else back by a decade, since no other nation could afford to try out so many methods in parallel.
Secret number 2: How much fissile material is needed? Finding the critical mass of enriched Uranium or Plutonium was really the key to making an atomic bomb or nuclear reactor work. Since there was no stockpile of enriched Uranium and Plutonium isn't a naturally occurring element, this complicated matters considerably. In fact, the actual nature of Plutonium could only be guessed at until sufficient quantities could be extracted from reactors. This actually provides an entry into:
Secret number 3: The layout of the bomb. While in general terms we can now describe how uranium "gun" bombs and Plutonium implosion bombs work, the actual mechanics are still closely guarded secrets. To use a simple example, many readers here can describe how steam, internal combustion or Diesel engines work, but how many of us could actually build one?
Nuclear weapons have other factors beyond what simple mechanical devices have to contend with. Consider, for example the "Thin Man" nuclear weapon. A relatively straightforward "gun" type bomb using Plutonium as the fissile material, the design could not be made to work since the Plutonium would spontaneously fission before the two parts of the mechanism coldcould be brought together (even at 900m/s). This is because nuclear reactions are ridiculously fast compared to mechanical and chemical reactions.
If you really want an "alt history" where the mechanism for nuclear weapons remains secret, have a history where there are no Soviet spies in the nuclear program, and the US successfully implies that the working nuclear devices are all derivatives of the "Thin Man" (Dummy bomb casings shaped in the distinctive long, thin cylinders of ultra high speed gun bombs are prominently displayed, for example).
Since everyone knows that nuclear reactions are possible, and can eventually discover how to create and separate Plutonium, they will still be going on a wild goose chase trying to determine how the Americans are making their bombs work while Russian, French, Chinese and other bombs do not....