The discovery of nuclear fission in uranium, and its possibilities for releasing energy, were widely reported in major international newspapers. People who kept abreast of scientific news would have known about it, in the same way that people who keep abreast of scientific matters today know about, say, CRISPR. There would be (as today) a lot of people who don't know what that is, but there would also be a lot of non-scientists who did know about it, and knew that it was heralded as having great potential and possibly danger.

The idea of "atomic bombs," broadly speaking, had been around in science fiction circles since the 1910s and the discovery of radioactivity. The notion that "atomic energy" had lots of locked-away energy was also very common.

To use another modern analogy, the level of popular recognition would be on par with the idea of a warp drive. Everyone pretty much knows what a warp drive is supposed to be (a spaceship transportation method that somehow goes faster than light). Some people know there have been speculations as to how to do such a thing in real life, but also probably assume that one isn't going to be built anytime soon. If tomorrow it was announced that a warp drive had been created and that people had visited Pluto last week, well, you'd be pretty shocked, but the underlying concepts would probably be at least somewhat familiar to you. Similarly with the atomic bomb — nobody really expected one to be built and used during World War II (there had been rumors but they were easy to dismiss), and to suddenly have a working version, seemingly out of nowhere, was a big shock, even if the idea was not totally alien.