Without the cold war to drive things along, I’m not sure that we ever would have gone to the moon.
Big rockets came out of weapons. If you are just tinkering for learning’s sake, you end up with things like Goddard’s early rockets. It took WWII to inspire the Germans to create the V2. If you don’t have a cold war and don’t have a need to send huge rockets carrying nukes over to the Soviets (and for them to do the same to us) then the military focus becomes short range missiles instead of the Saturn V. You aren’t getting anywhere near the moon on a short range missile.
There are plenty of uses for satellites, so I’m sure those would have been developed with or without a cold war, eventually. And once you have a rocket capable of lifting a satellite then you start reaching for the moon, but I think that everything would have been delayed enough that smaller robots would have been developed long before we sent human beings to the moon. We would probably be well into the 2000s before we managed to send a robot to the moon.
In this alternate reality, I’m not sure that we would have ever devoted enough money into the development of something like Skylab. Some Evel Knievel type daredevil would probably hold the high altitude record for some sort of rocket plane that would play in the same ballpark as the real-world X-15.
Getting to space is hard. No country has ever done it without a huge backing from their government, and even then only three countries (U.S., Soviet Union, and China) have sent their own crewed vessels into space. Only a handful of other countries even have the ability to put something into orbit.
Sure the U.S. has private space vehicles now, but that was built on the shoulders of the U.S. space program, and even though there are several “private” space companies now, those companies wouldn’t be in the space business if they hadn’t had some major help from the U.S. Government. NASA paid for over half of SpaceX’s rocket development, for example, which arguably is a good thing because SpaceX was able to provide solutions significantly cheaper than if NASA had done the work themselves. But the point is, without NASA, there would be no SpaceX.
The Space Shuttle did a lot of classified military work. And even SpaceX has military contracts funding at least part of their work.
If you take away the military funding, everything goes away. There isn’t enough money left over for a serious space program of any sort.