I know it would be possible that our former Cold War enemies would have achieved their space dreams had they fixed many of their technological mistakes. My possibilities....
Had Sergei Korolyov never died in the failed surgery in 1966, his Soviet space program would have continued not just his survival but perhaps the resolution of between himself and rival scientist Chelomei by sharing the same data and results that would have led the success of their lunar program. Therefore the Russians would have won the Space Race by landing their first man on the Moon by 1969.
The reasons why Russia never sent their astronauts (or cosmonauts, the Russian equivalent) to the Moon through American victory is not just because of the downfall of the lunar program after Korolyov's death. Here they are:
1.) First, the program's funding overlapped many areas of research and engineering. The rockets that sent Gagarin and later cosmonauts to space, for example, were actually modified military missiles, or ICBMs.
2.) Second, rival groups had lack of focus and political in-fighting. They had the same goal of garnering the glory for themselves but never had its share with others. As I've mentioned earlier, Korolyov and Chelomei had the same aim of reaching the Moon and gathered their own data and results, but they never co-operated.
3.) Third, space programs were kept state secret. While the United States shared their scientific data to the public due to their freedom of the press, the Soviets only kept their latest technological projects, especially the latest lunar program, out of the public.
4.) Fourth, space projects that were not launched were canceled for political reasons. High levels of risk were taken, so there were several accidents and deaths that put Soviet space projects on hold or cancellation, including a few botched secret test launches of the giant N-1 booster that exploded during liftoff.
Back to the (would-have-been) Soviet Moon landing: The first cosmonaut's lunar steps would lead later lunar missions as a preparation for future lunar colonization. Salyut space stations would also develop space capsules and lunar rovers to the Moon, where the first moon base would be constructed by the end of the 1970s.
Meanwhile, for the US, their lunar missions would be put on hold. Future manned missions such as Skylab station and the space shuttles would either go on, be postponed indefinitely, or be officially terminated forever, leaving unmanned missions such as the Viking and Voyager space probes to continue as planned. Many of the funds for the manned programs would end up for either military uses or social programs (i.e., health care, education).