There is a section in the FAQ about this but the TLDR; answer is, "not close." They were not trying to develop an atomic bomb; they had a very modest nuclear reactor pilot program. The idea that they were actually trying to build a bomb during the war is one of many myths surrounding World War II.

It is impossible to guess when they would have developed an atomic bomb had World War II not been going on, but for the sake of comparison the Germans in May 1945 were not quite to the stage that the American nuclear project was in December 1942 (they did not yet have a working nuclear reactor prototype, but were getting closer to that). The US Manhattan Project is still the fastest nuclear weapons-development program ever, so it puts some bounds on it (e.g., not likely before 1948, even if they invested a Manhattan Project-level of resources into it from that point onward).