I think one of the aspect which people often ignore in the space race are economics. The space race was pushed by having two big countries who was able and willing to put a lot money into space travel to brute force it while the technology was still immature. So the big question is to ask how do the economy look in alternative history.
I think a space race in CP win WWI would be interesting. In that would we could likely see a multi tier space race. Germany, AH and UK are big enough economies to compete with USA and USSR, while France and Japan will be strong second tier players. Germany and AH will likely integrate their programs to some extent, while UK without WWII will still deal with the decline of their empire but do so on more manageable basis [1] and as such still have the resources to throw in the space program.
[1] losing India, but likely keep a strong control over the Persian Gulf and its oil production, and have several Dominion (Canada, Australia, New Zealand, Singapore and Malaysia) stay economic and military integrated with the UK.