This is a very tricky challenge, since it depends so heavily on the precise balance of power during (and after) World War II, and the existence of another technology: nuclear weapons. The difficulty is that a state in the same relative position as the USSR IOTL (that is, a strong continental power but outmatched at sea and to some extent in the air, and with nuclear weapons) will face an irresistible pressure to develop ICBMs, since they completely negate the enemy's advantages, and once you have ICBMs space flight is a trivial propaganda coup, and besides extremely valuable militarily (communications, positioning, spying...).
So, the challenge comes down, essentially, to having a World War II (or early Cold War--pre-1949) where the Soviet Union is negated (although not necessarily totally destroyed) as a power, while Nazi Germany is not made strong enough by this fact to defeat or at least deter the WAllies. Even avoiding World War II entirely isn't enough, since the Soviet Union has as much or maybe more incentive to develop IRBMs and ICBMs in that scenario as IOTL.
This could, perhaps, be done by having the WAllies send less Lend-Lease aid to the USSR, causing them to advance more slowly and have more problems consolidating what gains they do make. Thus, by 1945, when the Americans and British are marching into Berlin, the Soviets have hardly finished retaking control of their pre-Barbarossa territories. Some crisis like what Blue Max says then erupts within 2-3 years, and the WAllies are forced into actually invading (or nuking) the USSR, causing it to partially collapse, or at any rate stop being a superpower, and no longer pursue a nuclear weapons program. The WAllies, with their bomber capability, do not need to develop ICBMs to effectively threaten other countries with nuclear bombing, and they don't need spy satellites to be able to effectively observe other countries (as they did IOTL, due to the development of Soviet SAMs and interceptor aircraft).
However, even under the most favorable conditions, I don't think you can prevent space flight past the 1970s. The problem is simply that it is very easy to launch (very) small satellites into orbit. A rocket like the Scout (introduced in 1960), the Lambda (Japanese, with the first orbital launch in 1970; it wasn't even guided!), or Pegasus (introduced in 1990, but the basic concept is similar to early rocket experimental aircraft) would not cost all that much to develop from existing large sounding rockets, and would offer at least some scientifically interesting capabilities. The difficulty would be comparable to a large particle collider or telescope, which as we see have been developed IOTL for purely scientific interests. There's no reason to suppose that this wouldn't happen ITTL for the same reasons. Even human flight is fairly likely, although much later than IOTL, once sufficiently large satellite launchers have been developed (which they probably will, after a few experiments show the potential of weather observation, communications, and remote observation for commercial and governmental uses).