Nuclear fission was discovered in December 1938. By early 1939 it had spread to scientific communities near and far, even before it was published, but definitely after that. That there was a theoretical possibility of using it in a reactor or weapon was realized by numerous scientists in early 1939, but it was far from clear that you could actually do it practically, or at least anytime soon. But fairly soon after the announcement of "secondary neutrons" by Joliot in the spring of 1939 (which enable a chain reaction to be possible, making at least reactors seem feasible), scientists in a wide number of industrial nations (the US, UK, USSR, France, Germany, and Japan, among others) petitioned their governments for funding to look into the matter.
Government funding and interest was initially somewhat muted. These were theoretical possibilities, and a big war was getting underway. Even the scientists who made these initial petitions did not promise any miracles. Today we read the Einstein-Szilard letter, for example, as the start of the US atomic bomb project, but if you read it you will find it both quite cautious in its diagnosis of feasibility (bombs may be possible) and very restrictive in its recommendations (it recommends creating a liaison who would help fund experimental work in labs and keep track of the research — that is not the same thing as saying "build a bomb and use it").
Work on these matters continued largely in university laboratories with some government support in various nations through the beginning of World War II. Most of the work being done was "exploratory": it was of the character of figuring out whether bombs or reactors were possible, not actually trying to build bombs or reactors. In only a few countries did this work advance to a "pilot" stage, where they actually tried to set up prototype reactors. In only one country did it advance to a "production" phase where they actually were trying to build a bomb.
The German and American cases are worth considering. Both were more or less at the same level of investment and interest until 1941 — both had "exploratory" programs. But they diverged. The German scientists recommended moving into a pilot phase, but for reactors. To imagine creating bombs would require a massive resource investment the German scientists were unwilling to make given their lack of certainty in the success of the project. In other words, the amount of risk was high and it looked like a very difficult job. So they instead had a very modest pilot reactor program instead, and while they made significant progress despite the war, they did not quite get even a prototype reactor running by the time it ended for them.
In the US the situation is notably very different. The Einstein letter of 1939 set up an exploratory committee. This did not make terrible amounts of progress for a variety of reasons. By 1941 or so the top US scientists were pessimistic about nuclear fission making any kind of war contribution. But then an intervention came from outside: a similar exploratory committee in the UK (the MAUD Committee) had concluded that in fact there were probably reasonably good odds at making a bomb, and that Germany was probably trying to make one. In retrospect they were both wrong on Germany and wrong on the odds — it was much harder than they had expected to pull it off. But they managed to convince the top US scientists that this was a real possibility, and these scientists quickly put into place a pilot program (1941-1942) and then got approval by late 1942 to start a production program (the formal start of the actual Manhattan Project). Again as noted this was harder than they expected (it cost 4X more than they budgeted originally), and they barely managed to pull it off (they managed to have three bombs ready by July 1945 — a few more months later and they might not have been usable before the war ended), but they did pull it off.
So you can see here how the technical side of these things is only part of it; the rest are judgments about feasibility, risk, and an ability and willingness to persuade politicians and administrators to support such a project. That only occurred in the USA.
The standard and still-good source of discussion about scientists and the bomb is Richard Rhodes, The Making of the Atomic Bomb (1986). There is also a nice article on scientific reactions to the discovery of fission: Lawrence Badash, Elizabeth Hodes, Adolph Tiddens, "Nuclear Fission: Reaction to the Discovery in 1939," Proceedings of the American Philosophical Society 130, No. 2 (June 1986), 196-231. For a broader discussion of lots of nations, I believe that Helge Kragh's Quantum Generations, chapter 18, covers this in a somewhat more comparative light. On the German case, see Mark Walker, Nazi Science: Myth, Truth, and the German Atomic Bomb, which is a readable overview (he also has a much more technical book if you are so inclined).
