The idea of an "atomic bomb" was already a staple of science fiction and popular culture, and had been since the 1910s. The idea that nuclear fission might be able to produce one is the sort of thing that wonky, scientifically-interested Americans would know about, because it had been published in several major newspapers in the late 1930s and early 1940s. The idea that the United States was engaged in building an atomic fission bomb was a secret, and would have been known by nearly none.

Here is what I wrote recently on this:

The trope of releasing nuclear energy became very popular in the early 20th century, connected with the new science of radioactivity. Frederick Soddy, a chemist who worked with Ernest Rutherford, was responsible for a lot of the early atomic hype, with popular books like The Interpretation of Radium (first edn. 1909). This is where you start getting quips about how the energy in a glass of water could let you go around the world X number of times in an ocean-liner, etc. The first discussions of "atomic bombs" were in H.G. Wells' The World Set Free (1914), though they were not, obviously, fission weapons (they were more like bombs that caused matter to disintegrate, which caused any other matter it touched to disintegrate).

Even Winston Churchill got into the atomic hype in 1925, giving a speech which asked, "Might a bomb no bigger than an orange be found to possess a secret power to destroy a whole block of buildings — nay to concentrate the force of a thousand tons of cordite and blast a township at a stroke?"

By the 1940s this was all standard sci-fi fare, and the discovery of fission did fuel this speculation. But most scientists thought this was probably at decade from being realized, if it could be realized at all. Dreams of releasing endless energy had been dashed before — the atom did contain a lot of energy in it, but getting it out, in quantity, was the hard part.

The analogy I like to use is that of the "warp drive" today. We've all heard of such a thing, we all know more or less what it might mean (faster than light travel, Star Trek, etc.). The geeks among us know that there are hypothetical means of maybe making it work (e.g. the Alcubierre drive), but nobody but a crank would think they are just around the corner. Now imagine that President Obama announced tomorrow that a warp drive had been developed and gone around the solar system ten times as a result — you'd be shocked. But you would recognize the term "warp drive," even if you had no idea how they had actually pulled it off. This is approximately what the public understanding of the term "atomic bomb" would be in 1945 — the idea of atomic energy was known, the idea of making a bomb out of it was known, the specifics were not known.