To call it an alliance stretches that word almost to the point of nonrecognition, IMHO. I tend to think of it as more of a preemptive ceasefire between two enemies in order to first finish off a common enemy.
I wonder what would have happened if we had dropped the bombs on Japan before Germany surrendered rather than after. I rather think that Stalin would effectively drop the agreements and push his troops to the point of collapse in order to take as much of Europe as possible, beyond what he got, in order to deny the US ground in Europe from which it could threaten to drop nukes on the USSR. As it was, by the time we nuked Japan the facts on the ground in Germany were sufficiently set that Stalin couldn't really risk trying to restart the offensive. Aside from the logistics issues, he would have had the risk that the US had more nukes and would have used them on the Soviet Union itself if he ramped up his machine again and repudiated the terms of surrender.
We were out of bombs, yes, and it was 11 months until the first tests at Bikini. That said, we'd have had more a lot sooner if there was still a shooting war going on.
This seems like a good place to mention, if both the US and Russia are going to be making airstrikes in the same area, we should at the very least set up some sort of mutually agreed upon air traffic control system, or it's only a matter of time before somebody shoots down the wrong target.