The major difficulty with this is that the Soviet Union was not at all prepared to build the atomic bomb. At the start of the war many Soviet scientists believed that nuclear research would not yield any practical results in the near future, and this was reflected by how low of a priority the state gave the atomic bomb project until the Hiroshima and Nagasaki. At the start of the program the Soviets lacked basic materials like heavy water and uranium.
The best way to fix all of this is to find a way to get the Soviet leadership to make the atomic bomb a priority. The problem is that I can't think of any motivating factor short of another nation testing the bomb (like OTL).