Avoid WWII, and Europe, plus Japan and China, are much richer and more industrialized than IOTL. Millions of potential engineers and scientists killed IOTL are also alive. They are also in an arms race against one another, meaning that many more nations work on building missiles and atomic weapons instead of MiG clones.
By the 1980s, I could reasonably see a Japanese, German, British, French, Chinese, and possibly even Czech and Polish and Romanian space program (gotta have spy sats to watch the Soviets). Most of those would be unmanned, or only manned as part of some ESA-like joint effort, but they'd stimulate further growth. EDIT: But the bigger European and Asian economies, at least, could probably afford a manned program. The second-tier powers might do it as a joint thing--'Little Entente in Space.'
EDIT2: as for precise POD, have Germany lose WWI harder, so that there is no question of revanchism, and have Poland and Czechoslovakia come to a more agreeable solution for Cieszyn so that they work together to contain Germany down the line.