Answer:
No, Truman didn't directly order either the bombing of Hiroshima or Nagasaki.
What Truman approved was an order to use the first bomb (the "Little Boy" uranium atomic bomb, on or about August 3, 1945, and "Additional bombs will be delivered on the above targets [AW: Hiroshima, Kokura, Niigata and Nagasaki] as soon as made ready by the project staff."
So Truman wasn't dropping an atomic bomb and waiting to see if Japanese leaders surrendered, repeat as necessary. Truman authorized the use of the first two atomic bombs as soon as they were ready.
Truman didn't intervene to override these orders until August 14, 1945, when he ordered that the third atomic bomb being readied on Tinian for use around August 19 only be used on his explicit orders.
So was Truman influenced by the impending Soviet declaration of war against Japan and invasion of Manchuria? Yes, Truman was.
While at Potsdam, Truman received word of the successful test of the Fat Man atomic and a prospective schedule for their use on Japan. The Little Boy uranium bomb was being assembled on Tinian and would be ready around August 3 - 4. The Fat Man plutonium bomb was also being assembled but would be ready later, thanks to its more complicated design.
The orders for use of the atomic bombs required that the bombardier have visual sighting of the target to drop the atomic bomb. Forecasters had predicted about a week of clear weather around August 3 for the atomic bombing missions. Truman wanted the atomic bombs used before the Soviet Union entered the war against Japan and it looked like he had a narrow window to accomplish it.[1]
Meanwhile, Stalin was planning his endgame against Japan.
When Stalin ordered the Red Army to begin planning for the invasion of Manchuria, he set a date of about August 25 for the invasion. In July, shortly before the Potsdam Conference, Stalin moved the invasion date up to August 15.
During the Potsdam Conference, Stalin informed Truman of the August 15 declaration of war / invasion of Manchuria date, but after Truman had informed Stalin that the US had developed an atomic bomb, Stalin moved up the invasion to August 9. Unlike Stalin, Truman did not confide a date when the atomic bombs would be dropped on Japan.
So Stalin wanted his declaration of war and invasion of Manchuria to happen before the US had a chance to use the atomic bombs and Truman wanted to use the atomic bombs before Stalin entered the war against Japan. Truman edged Stalin at the tape.
[1] A lot has been made of Truman's motives about using the atomic bombs before the Soviet invasion. Some historians such as Gar Alperovitz have argued that Truman intended to intimidate Stalin with their use. There's little evidence that Truman wanted to intimidate Stalin, in fact, Truman's personal diaries from Potsdam imply that Truman thought that both, two atomic bombs and a Soviet declaration of war, would be necessary to induce Japanese leaders to surrender. In August 1945, Truman still considered Stalin and the Soviet Union as an ally and a necessary player in the post-WWII world.
