When did Truman make his choice to drop the atomic bomb?

Answer 1

Actually Truman never made that choice, Franklin D. Roosevelt did in the last few months of his life when it became obvious that Germany would surrender before the atomic bomb could be available but Japan would likely "fight to the last man".

Truman simply decided not to change any of Franklin D. Roosevelt's plans which the military was in the process of preparing the final planning for their execution. These preparations included: 2 D-day invasions of Japan (each larger than the 1944 Normandy D-day invasion), stockpiling of chemical weapons (e.g. nerve gas, mustard gas) for use during the invasions, continuation of the multiple nightly 1000 plane firebombing raids, use of atomic bombs "as they become available".

Following the successful Trinity test on July 16, 1945 the Manhattan Project was able to brief Truman that they should be able to produce and drop 23 atomic bombs on Japan before the end of 1945 (this was enabled by the more efficient plutonium bomb design, had the test failed the uranium bomb design would have to be used and it would have been unlikely that more than about 6 atomic bombs could have been produced and used before the end of 1945). This reassured Truman enough that he mentioned that the US now possessed a powerful new weapon to Stalin, he was a little surprised at Stalin's lack of response (but as we now know Stalin had already heard of the results of the Trinity test and likely knew more about it than Truman did).

As everything had already been authorized back in May 1945 when the target list had been prepared and the orders to "use the bombs as soon as they become available" were written, Truman's only real choice was to stop the use of the atomic bombs and he made this choice as soon as he received the Japanese message of surrender on August 15, 1945.