As long as the US has the bomb, the USSR - and the UK - has to have it. The only two ways I can see to prevent Russia from getting it are if either the US launches a preemptive war or the Great Powers reach an agreement to outlaw atomic weapons. Both options are extremely improbable, but - at least for item two - I don't think they are absolutely impossible. However, as long as the USSR exists and someone other than Russia has an atomic weapon, Russia will need to have one too. So the US being sole possessor is pretty much ASB.
That said, taking the OP as writ, let's work through this.
I don't think it changes much right away. The USSR didn't have enough bombs during the late 40s to make a difference and that didn't keep the Korean War from happening, so I think it probably still happens. And Truman isn't going to drop the bomb for anything short of an invasion of western Europe. So I think Korea still happens as per usual. This might have significant impacts on the Red Scare, although I'm not sure exactly what. Development of the H-bomb is probably delayed, possibly for quite some time.
I think it's in the 50s we start to see significant divergence. No Soviet ICBMs, for one thing, and probably very delayed American ones - maybe they don't even happen at all. That puts a serious crimp on the space race and satellite reconnaissance. The USSR probably starts to seem a lot less threatening, especially once Stalin is gone and Khrushchev doesn't have a rocket to rattle.
The key question is how the US reacts to discovering it is, for all practical purposes, the preeminent military superpower. Do we basically do what we did IOTL, get a better night's sleep? Or does containment get replaced by rollback, and we start getting into even more unpleasant stuff than IOTL?
Similarly, what do the Soviets do in reaction to their evident inability to replicate American technology? People always immediately say "bioweapons", and that's certainly a possibility, but I wonder what other effects it would have on their politics.