My suggested checklist:
1. Delay US satellite launching program. This will be easier if you
2. Delay USSR satellite launch. So, kill off the ICBM (or its designers) in some way. Perhaps Korolev has an accident in his Gulag.
3. Bring the UK ICBM program forward. Have Sandys get in earlier and axe all of the pretty ships and planes, and put Great Britain on a fast-track to big rockets.
4. Keep the Fourth Republic in France, along with its cooperation with Great Britain. A good way to do this would to
5. Have a different Suez. Eisenhower was quite the meanie to the Europeans, and pretty much killed the idea of looking to the UK and France as significant international players on the order of the two superpowers. Perhaps if someone who was a lot more careful about upsetting Cold War Allies was president the US would just shrug and tell the USSR "a nuke for a nuke"...so how about
6. Eisenhower has a stroke in 1955 and dies, leaving Nixon in charge, or isolationist/anti-communist Taft wins the nomination in 1952.
So now you have a situation where the US is more isolationist/pro-colonial up to 1960, Britain and France are seen as poles around which a European team can realistically be built (although everyone will know who is in charge!), and Britain and France jointly lead the world in rocketry development in an attempt to "more cheaply" toss nukes at Moscow.
That's my best shot...in text form:
Korolev dies in a Gulag pre-WW2. Flash forward to 1952, where Bob Taft wins the nomination, picks Richard Nixon as his veep, and then dies after a month in office.
Duncan Sandys gets placed as Minister of Defence in 1954, rather than spending three years at Housing, so we get a 1954 White Paper that says that "Rockets are the Future!" (This, incidentally, will kill the Victor bomber, so you get 2/3 of the "V-Bombers".) Assuming that missile development takes four years with joint French funding and project fast-tracking, and assuming two years of very intensive testing before a payload gets launched (judging from the attempts at Woomera with the ELDO after the cancellation of Blue Streak), you get a Euro-launch of a payload in 1960.
As for the aftermath, the US and USSR will be right behind the Europeans, the RAF bomber force and the RN will be gutted, and the Euro-organization will topple before a manned flight because it's just too damn expensive for the UK and France to play at being superpowers with space shots when those missiles need to be saved as the very expensive deterrent they were always meant to be. But you will have your ten to twenty weeks of glory in 1960, and plausibly to boot.
Hope that helps!