The USSR was first in space, weren't they? The technological advances of the USSR can't all be stolen unless they first invented time machines.
So no need for any time machines :)
There are a few good books on what Korolevs teams did which are pretty fascinating in hindsight, especially due to limited access to computers in USSR. (Obviously sources about the US space programs are about 100x more numerous in the western world :) )
The soviets did start by copying the V2, but immediately after that switched to original designs.