The Rosenberg spy ring was in fact pretty incidental to the Soviets getting the bomb, as an aside. Even General Groves, head of the Manhattan Project, admitted the information they gave up was pretty "minor" (off the record, of course):

I think the data that went out in the case of the Rosenbergs was of minor value. I would never say that publicly. Again that is something while it is not secret, I think should be kept very quiet, because irrespective of the value of that in the overall picture, the Rosenbergs deserved to hang, and I would not like to see anything that would make people say General Groves thinks they didn’t do much damage after all.

But there were others (like Fuchs) who gave more useful information. However, we now know, looking closely at Soviet archives, that they did not use that information as efficiently as one might imagine they would have in the abstract. The head of the project, NKVD chief Lavrenty Beria, did not trust espionage information in an unqualified manner. (Given how "double-cross" operations were key to counterintelligence during World War II, it is not a paranoid thing to do, to distrust people abroad who claim to be giving you secret information.) Instead, he had the information processed to be made to look like it came from another Soviet laboratory, and used it as a "check" and only somewhat as a "guide" for his scientists (who he also did not trust). Which is to say, it played a far more peripheral role in the Soviet program than people tend to think — every aspect of the data was re-checked, and all the parts indicated as "blind alleys" were still investigated (because even if the spy data was genuine, it could still be wrong). It is pretty clear that the timetable of the Soviet program was set not by information acquisition, but by acquisition of raw material (uranium resources) and the production of fissile material (the fuel for the bomb). Many of the details of this sort of thing were not (and could not) have been provided by espionage — most of the espionage information was about bomb design, because that is what the spies primarily worked on.

Separately, it is worth noting that after the war ended, there was plenty of useful information available to the Soviets that was not espionage in nature, such as the Smyth Report and other publications released about the wartime project. These were used to guide the broad outlines of actual development project (which did not start until the war ended), and used to narrow the avenues of approach (e.g., they knew reactors would work and that plutonium was usable in a weapon — that was something the Americans did not know when they started their work). This would have been available to them even without the spies.

Would the Soviets have been able to develop an atomic bomb without spy information? Without a doubt. They had a large industrial and technical base, and people of great quality working for them with nearly unlimited resources. Would their first bomb have looked different tan the one that was detonated (which was a Sovietized "clone" of the Trinity device)? No doubt — in fact, the Soviet scientists by 1949 had already developed more efficient bomb designs, and the 1945 version was only used because Beria wanted success to be guaranteed (because he feared Stalin's wrath).

There is literally no reason to think the Soviets wouldn't have been able to pull it off without espionage. As it was, the espionage played a much more minor and specific role than people think. The real question to ask is, would the lack of espionage have changed how fast they got a bomb? Which is the flip side of the question of how much time the espionage "saved" them. There is no real way to know this (it is a counterfactual), but given what we know about the Soviet project, scholars have estimated that at most the espionage information might have saved them six months to a year of time. I think even that is probably uncertain; I am not sure it really saved them any appreciable time.

Michael Gordin's Red Cloud at Dawn is a great resource on the Soviet bomb program and its relationship to Western information flows (both espionage and not). David Holloway's Stalin and the Bomb is a more detailed look at the Soviet program itself, albeit without as much access to Soviet records on espionage information (and its use) as Gordin's book (because a lot of those weren't released until relatively recently).