Slon, you can't just shrug off the bomb so easily.
USA had 3 fully prepared bombs ready for operation crossroads in July 1946, any continuation of the immediate conflict in '45 would have accelerated, and increased commitment to, bomb production. Refinements not required, anything nuclear that went 'bang' would be considered acceptable and even one device in the right place could be catastrophic for any target.

The Soviets would not have been exactly sanguine about the possibility of their use either (in a world where 2 major cities had recently been levelled).
It's important to consider they would be unaware of the true status of the US program, and as 40 years of cold war proved the threat of the bomb can be just as powerful as the actuality.

Cheers,
Adam.

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