NERVA's no good for getting from surface to orbit-its thrust is too low to actually lift itself off the ground. As such, even if we had NERVA, without some other change it won't have much effect: the bottleneck in space transport isn't moving around once you're in space, it's getting there in the first place. Being optimistic, maybe we'd have gotten something like the OTV, only nuclear-powered. That would get you some really sweet unmanned missions, and it would put us in a better position to send manned missions to different places, but without further changes (i.e., more money), it won't make much difference to the manned program.
There was a related program, called DUMBO, that had a hybrid nuclear-chemical system that was high-thrust enough for a launch vehicle. It was cancelled, allegedly because NASA knew that getting a nuclear space drive would be hard enough without trying to run it inside the atmosphere. I don't know much about the project, I'm afraid, so I don't know if it was at all viable or cost-effective. But if you want to get a Heinlein-esque Space Program out of an ATL without ASBs, nuclear launch is the way to do it. Chemical rockets don't have the exhaust velocity, even today; Orion is insane; and all the other interesting possibilities, like space elevators, require tech that we don't have even now.
(Except for maybe beamed power, but I don't know anything about that, so I can't say).