I don't think you're right on this one.
We've got a lot of uranium in Utah and Colorado, and, as I recall, the production schedule on atomic bombs was roughly one every two months after the Trinity test bomb, Little Boy and Fat Man.
I can't find a definitive reference on that one; I'm writing from memory. However, I find it hard to believe that we could crank out three bombs by 1945 and then just shut down production for four years.
As for battlefield use, it would have been impossible to reach any major soviet city, considering that from the closest bases it would have been a flight of at least a thousand kilometers across enemy dominated air space into the mouth of one of the thickest air-defense nets in the world. As for battlefield use, the range on the atomics was so small (radius) that they would have made little impact on a front line compared to the cost of making the bomb.