As many posters have pointed out, the knowledge of nuclear physics was simply too widespread by 1930 to really obscure the idea of a nuclear chain reaction. Atomic energy was so well known that H.G. Wells postulated a form of atomic weapon for his novel "The World Set Free" in 1914, although today we would recognize this as a "salted" or "dirty" bomb.
By the time WWII is actually under way, all the major combatants have some form of nuclear program. Imperial Japan, despite its lack of industrial capacity, actually has two programs, one conducted by the Imperial Army, and a totally separate one by the Imperial Navy.
However, the biggest problem in any nuclear program is actually getting the fissionable material. The Germans seemed intent on using natural Uranium moderated by heavy water (D2O), a process similar to a modern CANDU reactor. From the various books and documents that I have been able to read over the years, it seems fairly clear that few people had any real understanding of the difficulty needed to get fissionable uranium, and fewer still had the resource base to do so.
Only the United States had the resources and ability to do so, and even then, the Manhattan project needed to explore every possible means of enriching Uranium in order to discover which one was most suitable for industrial production of enriched Uranium for reactor fuel and bomb material.
Since none of the other Allied or Axis powers seemed to have the ability to find and scale a method of Uranium enrichment, then having either the United States sit out the War, or enter a year or two later means that only very tiny amounts of enriched Uranium are available for experiments, but never enough for power reactors or bombs. This prevents Uranium "Gun" type bombs like Little Boy, but the understanding of nuclear fission exists, and the production of Plutonium is possible, even with natural Uranium reactors.
This presents a problem, since even with 1940 era technology, the various special timers, switches, detonators and mathematical and machine tools needed to create a functioning "implosion" device exist (and this was demonstrated by "Trinity" and "Fat Man", the first and third nuclear weapons ever detonated).
Here a slight hand wave could exist. If you disallow the precision timers which create the conditions for the symmetrical implosion waves that make plutonium fission bombs possible, then the various powers will try to create Plutonium "Gun" bombs. However, this configuration is impossible, since the Plutonium will start fissioning under the presence of emitted neutrons even as the two pieces are being propelled together, creating a "fizzle" yield rather than a nuclear explosion.
Plutonium "Thin Man" bomb casings. At over 17' long, they were only able to be fitted in specially modified B-29's. Even a 3000' second "shot" speed was too slow to prevent uncontrolled spontanious fission
So essentially, all that can be done by the 1930's is to slow down the ability to create fissionable materials, and to handwave the ability to build the intricate timing and switching devices needed for implosion bombs away.