There is no reason to think that the USSR would not have been able to develop nuclear weapons without espionage. They had a well-developed scientific and industrial base by the end of WWII, they had many world-experts in the relevant subjects, they had access to scientists, engineers, and material from the German atomic program, they had a will and drive to complete it, and they had access to ample amounts of cheap (slave/prisoner) labor.

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.

The impossibility of answering this question conclusively is that it is ultimately a counterfactual: when one asks, "how important was the espionage?" one is asking "what would history have been like if they didn't have the espionage?" and that is something we cannot answer.

Lastly, I would just note that the places where the Rosenberg network made a real difference are not the atomic bomb. Their contributions there were pretty meager, and very minor compared to other atomic spies not connected to the Rosenbergs. The Rosenberg network's real contributions were in less-sexy but still-important military technology, notably the SCR-584 radar, M9 predictor, and proximity fuze (the latter of which Rosenberg managed to compile a full, assembled, working model for his spymaster), the combination of which is an incredibly powerful (for WWII-era) anti-air defense system. Rosenberg's network also gave huge amounts of information on the AN/APS-12 fire control radar, AN/APS-2 airborne microwave radar, AN-CRT-4 sonobuoy, AN/APS-1 airborne search and detection radar, AN/APQ-7 airborne radar, Westinghouse 19A jet engine, and P-80 Shooting Star engine. Rosenberg acquired so many materials in electronics work that he was rented special facilities for photographing them, because it ended up being thousands and thousands of pages. The atomic espionage they gave was paltry by comparison, and served primarily to just help confirm that the information from other spies (like Klaus Fuchs and Ted Hall) was not disinformation.

On the Soviet use of espionage information and the Smyth Report, see Michael Gordin's Red Cloud at Dawn, which is all about this. The most vivid account of Rosenberg's work is Alexander Feklisov's The Man Behind the Rosenbergs, which needs to be taken with a grain of salt (Feklisov was their spymaster!) but seems generally plausible to me (and is not strongly contradicted by other sources).