The basic idea that you might be able to take advantage of the massive hidden energies from radioactive decays circulated in the early 20th century, well before the discovery of nuclear fission. The discovery of nuclear fission proposed a mechanism for how you would manage that release, through the exponential chain reaction, where the splitting of heavy atoms released neutrons that would create more splitting. This was realized by many scientists, with the Hungarian Leo Szilard being the apparent first, around 1939, almost immediately after fission was discovered.

The fact that this discovery coincided with the beginnings of World War II encouraged further work in the US and UK. Many of the scientists who did the most to push for further research, investigation, and funding on this topic were Jewish refugees from Europe, and they were fueled by a fear of Nazi possession of nuclear weapons. Without these refugees it is unlikely that the atomic bomb would have been developed during World War II; the "native" UK and US scientists were by themselves not nearly as motivated in the early stages, because there were so many uncertainties involved that it seemed like a foolish thing to invest in otherwise, given the many other demands for scientific and industrial attention during the war.

As for the work itself, see here for a basic timeline of development. It was not rapid. From 1939-1941, the US did very little more than basic research in this subject. By mid-to-late 1941, they started much more serious work on the issue and had a much better understanding of what a bomb would look like. In 1942, they started ramping up pilot studies. By the end of 1942, they had committed to building an actual bomb and spending billions on it. This resulted in them having three weapons by the summer of 1945. So the whole process took maybe 6 years, though the real work to make the bomb was only about 2.5 years. The war definitely sped all of this up; normally the route from "basic science discovery" to "entirely new industry" is at least a decade long, often longer.

Richard Rhodes' The Making of the Atomic Bomb is the standard book on this subject, and it still holds up pretty well. Cameron Reed's The History and Science of the Manhattan Project is also quite good at charting exactly when decisions were made and why, and what their different scientific understandings were.