chapotraphouse

I hate to quote reddit but this post on r/askhistorian answered by Alex Wellerstein, a historian who wrote a book on the US nuclear secrecy, had a detailed answer:

The question is always, how much time was saved by the espionage? And that's not something that's easy to answer. They did not just take the espionage information and do what it said to do. Aside from the fact that most the work of the bomb did not involve the kind of things they got espionage on — they mostly got espionage on warhead design, and that is only one tiny part of the overall difficulty of making an atomic bomb — they also did not use the espionage information in a way that maximized efficiency and time. They used it in a way which maximized their confidence that they were going in the right direction, which is not the same thing, because the heads of the program didn't entirely trust either the espionage sources or their own scientists. So the espionage information became a directional guide and "check" more than a strict plan to copy.

They also had sources of information other than espionage that frequently get overlooked. The Smyth Report, for example, was an official history of the Manhattan Project that was developed in parallel with it, and released a few days after the Nagasaki bombing. It contained no "secrets" of the sort that the espionage was concerned with — e.g., details of bomb designs — but it contained exactly the kind of information you'd want to know if you were planning a large-scale project: what sorts of facilities were developed, what methods for making the bomb fuel and other necessary materials proved to work, what the general size and scope of the operations were, how all of those things translated into actual bombs as a timeline. The Soviet atomic bomb project was very explicitly modeled on the Manhattan Project as reflected in the Smyth Report.

The single activity that set the "timeline" for the Soviet bomb project most specifically was the acquisition of raw uranium resources and the turning of it into fuel. Not what we think of as the "scientific" work behind developing the warhead itself (e.g., determining the amount of fuel needed for the weapon, determining how to make it react). In fact, the Soviet scientists (who by and large were not even aware there was espionage information; what info they got was "sanitized" and filtered to look like it came from other Soviet labs) came up with more advanced weapon designs than the 1945 US ones, but were prevented from using them for the first test because the head of the project (Lavrenty Beria, Stalin's spymaster) wanted the first test to be a guaranteed success.

They also didn't use the espionage to "cut corners." If the espionage suggested that a value was X, they checked if it was really X. If it said, method Y doesn't work, they still looked into method Y. Both because they didn't trust the information entirely (Beria was not a trusting fellow) and because it's possible the US erred in some places (which, in fact, it did — the US concluded that the gas centrifuge method of enrichment couldn't really be made to work well, whereas the Soviets, using a team of German-Austrian engineers, managed to get it working, and the modern Zippe-type centrifuge is descended from the Soviet method, not the US approach).

All of which, to me, suggests that they would have been fine without the espionage information, and that you are not talking about a big difference in time to completion with or without it. The Soviet scientists were also able to keep close base with the development of the hydrogen bomb, the successful version of which they developed independently of espionage information, which again to me points to the idea that they were not in any way "inferior" in this respect.