The American and the Soviet program were influenced by finding blueprints of the V-2 from captured factories, as well as capturing scientists. I know that the Soviet's heavily based their R1 ICBM on the V-2, but were they influenced after the production of the R1, or did the program rely more on modified Soviet designs and scientists?
Both the US and Soviet programs were essentially based on German rocketry in their early days, very explicitly using the V-2 as the basis of their work. The American Redstone missile was just a scaled-up V-2, and the Soviet R-7 was a clustered set of V-2-like rocket engines.
Both countries of course significantly diverged from the V-2 over time, but you could argue that all liquid-fueled rockets essentially point back to the fundamental concepts that were piloted by the V-2. Both countries also developed solid-fuel rockets that were fundamentally different from the V-2.
The countries both used their Germans differently and had different sorts of Germans. The Americans got many designers and top-level engineers (like von Braun), and they let them run things in the USA. The Soviets got many lower-level engineers, and the Soviets did not trust the Germans to outright run things. So the direct German influence was arguably more strongly felt in the US program (and you can see it in their much later rockets like the Saturn V).
In neither case was the work just copying — both countries required complete knowledge of technology in question to innovate with it, and even when they "copied" the V-2 directly (as they did with the A-2 and R-1 rockets) they were doing so adapting to local production requirements that were not identical to the original V-2 (thank goodness). There are also multiple ways to "riff" on the V-2; the US preferred scaling the whole thing up (one giant fuel tank, one giant oxidizer tank, like the Redstone), whereas the Soviets preferred a "clustered" model (e.g., four rockets strapped to a central rocket, like R-7).
Read "Red Star in Orbit" by James Oberg. It details the early days of the Soviet space program led by Sergei Korolev, a survivor of the Gulag arctic camps, sent there when Stalin purged the Tupelov aircraft design bureau. Rehabilitated under Khruschev, and for a while still technically a political prisoner, Korolev's iron will and genius forged the Soviet space progam.
NOT captured German scientists.
Can anyone answer how the Germans got to be so good at making rockets?