The big advance wasn't rockets, but guidance packages. After all, fireworks had been one of the early uses of gunpowder, more than half a millenium earlier. Without guidance packages, you have only the tiniest control where it'll go. A big rocket without guidance is just a random threat to civilians, not a help to anybody.
Guidance packages aren't the least bit easy. They still take alot of effort, even with today's computers, and Hamas' infamous rockets are unguided and rarely actually hit anybody or anything signicant. Google for Circular Error Probable, I think, will help you understand as well.
Guidance packages were just barely feasible in WW2 tech, notice, also the first generation that could just barely make computers. And, they could barely make guidance systems, either - Apparently, all the V1s and V2s were aimed at London Bridge, and not only did it stay standing, but they mostly managed to miss London. So, I'm dubious we'd see faster development. Now, you COULD have Goddard - maybe with some other, more politically oriented ally -get military funding and have and American military rocket program fun. It would, of course, tend to have similar limits to the V2.
Also, the first one of two of ANTHING new tends to be problematic.
There's another thread on a related topic, titled something like How I Came To Stop Worrying and Love the ICBM.'