You're right, Frank--fear of a German A-bomb put the Manhattan Project on a crisis footing. But, just before the end of the war, the US captured Werner Heisenberg and his team, and found that they hadn't even come close. They'd built a primitive reactor and it had never gone critical.

There were no "secrets" of nuclear fission after 1939--only technological issues, such as how much fissionable material was needed for a bomb, how to produce it, and how to trigger a chain reaction in a millionth of a second. Even before the first test, most Manhattan Project scientists predicted that another power--probably the Russians--would be able to build and test their own bomb within 3-5 years of a US success. It took them 4 years. They got help from spies, but all it did was to shave about 12-18 months off their effort by eliminating some trial and error.