Kennedy's original enthusiasm for the space program may have been as a way to show up the Russians. However, according to the 5 remaining Mercury astronauts in the 90's documentary Moonshot, Kennedy quickly became quite a fan of the space program. And these men don't strike me as the kind to accept any bull that is slung their way.
With all due respect to the Mercury 7, you could say that about a lot of the Presidents, most of whom were not noted (political) supporters of the space program. Nixon in particular is said to have really liked being around astronauts. But as far as space was concerned, his Presidency was an outright massacre. The documentary evidence shows a man who was greatly ambivalent about the Apollo program, and thought it was at best a secondary concern that he had been forced into by the Soviets.
I believe the Apollo project would have pretty much occurred as it did in our timeline.
Most likely something more or less comparable to the Apollo program happens, yes (the details may greatly differ, however; JFK in office may mean less "go fever" in '66 and so an avoidance of the Apollo 1 fire, or it may lead by-and-by to Apollo missions being canceled after 11 and 12 prove the point, or so on and so forth). What I was drawing attention to was the fact that it is
plausible that the Apollo program gets entirely derailed by détente kicking in and JFK making some kind of agreement on a joint mission with Brezhnev, say, or redefining the goal he set to just be about flying around the Moon, or similar scenarios.
With a workable team in place I think we might not have seen a permanent lunar base but maybe a permanent space station more like unto what we have now. It would have been harder to push the benefits of a lunar base at the time but a permanent space station would have all sorts of advantages in several different areas for all concerned.
A space station program was a plausible path forward for NASA after the Apollo program, yes (you might have noticed I helped write a little thing about what such a program might look like). However, it wasn't the
most likely program. See, the thing is that a lot of folks in NASA, by 1968 or so,
really wanted a space shuttle. Like
really really wanted a space shuttle. They figured that if they could get that into place then costs would drop a lot and they could get the space station
and the moonbase
and even a Mars mission down the line. Conversely, on Capitol Hill and in the White House folks were amenable to this argument because they wanted to cut costs a lot. They didn't want to give up the human program entirely, of course, but they definitely did not want to keep spending huge sums of money--and they
were huge, at the time--on sending people into space. Plus NASA argued that cutting costs so much would spur lots of private interest and would help save money all across the government, like with launching Department of Defense satellites. So here they had a program that would cost a reasonable amount of money, promised to save money in the future, and didn't commit them to indefinite high expenditures the way a moonbase or Mars program would.
The space station, by contrast, had a number of issues. First of all, it was a little more explicitly aligned with NASA's eternal ambition to go past LEO, which wasn't popular on the Hill. No one wanted to do that. Second, it just wasn't as...promising. Sure, you could do research there, have some diplomatic benefits, and so on and so forth, but the Space Shuttle offered to cut costs
dramatically and create entire new industries. That's way more enticing. Third, it didn't offer much
new to people. That is, new contracts that could be used for electoral advantage. Sure, there were the contracts for the station itself, but any realistic station program would mostly be launched on existing rockets using existing capsules. At most you might swing McDonnell Douglas to having a Big Gemini contract or
perhaps a new minishuttle. Small potatoes, really.
So the upshot is that you're just a lot more likely to get a space shuttle than a space station, even if you get the right people in place who realize that the fun times are over and they need to be thinking austerity and rational planning now, not dreaming of a red desert. This doesn't necessarily have to be
our space shuttle--there were still lots of ways to fiddle with that--but it's going to be
a space shuttle of some kind, probably.