“The Russians were also persuaded at Yalta to renounce their neutrality agreement with Japan, which they did in April 1945. This saved Allied lives by speeding up the Japanese surrender and capitulation.”
In the end, the bombing of Nagasaki compelled Emperor Hirohito to order his government to surrender whether or not Russia entered the war against Japan. So, the Russian war upon Japan saved little or no American lives, but it did set the stage for the Russians to take a large number of American lives when they planned and staged the Korean War only five years later with the participation of the Soviet air forces, air defense forces, intelligence groups, and assorted other Soviet units.
The Russians also imprisoned and liquidated the lives of more than about 36,000 British Commonwealth and about 24,000 American POWs taken into Soviet custody in the German POW camps in 1945.
Meanwhile having denounced their neutrality agreement in early April the Soviets declared war on Japan on August 8th, and were poised to attack their northernmost home island of Hokkaido in a matter of days. The Japanese simply did not have time to counter this threat, nor could they defend themselves simultaneously on two fronts.
Dropping atomic bombs on Hiroshima and Nagasaki may have provided a dramatic exclamation point to ending the war with Japan, but from a strictly military perspective wasn't necessary. The Japanese ran out of strategic options when the Soviet entered the war against them.