How about the opposite question? What if the Nazis and the other factors that led to WWII had been defused or dealt with quickly so that it never happened, or at least so that it never escalated into a World War? My opinion is that any modern space program would be even less advanced than what we have now.
I'm no historian and couldn't even guess at a timeline. But at the very least FDR probably wouldn't have been re-elected (and I'm not sure Churchill would have come to power at all). The U.S. probably wouldn't have pulled out of the Depression any time soon without the war driving it. There'd be no push to make a lot of the industrial and technological developments that were driven by the war, including nuclear weapons.
On the other hand, the rivalry between the U.S. and U.S.S.R. probably would have occurred regardless and might have resulted in full-scale war without the nuclear deterrence we had. Perhaps sooner or later that would have spurred a lot of the same developments anyway?
Opinions? Feel free to shoot down any of my points.
For a long time, I was flabbergasted at how much of the space age was pioneered during WWII. Now that I'm more used to it, I'm only continually amazed.