(This is definitely not a question for r/askphysicists. If you have a technical question, they are great. If you have a historical question, especially one about policy or politics, they are usually not the ones who have the best answers. These are very different types of expertise.)

So this question is an important one, but a difficult one to answer. In some sense, the answer is, "yes, Hiroshima and Nagasaki did lead the Soviet Union to develop nuclear weapons." Stalin accelerated the then-small Soviet nuclear program into a production program almost immediately afterwards, and was very explicit in talking to his scientists about their goal being to "not be another Japan." It is very clear that the bombings played into his thinking about future Soviet security needs.

But the tricky part of this question is that it is, to a degree, counterfactual. Whenever you add something like this, you are in very tenuous territory: "Can we assume if the project hadn't happened to begin with that no other similar project would have happened?" How can we know such things about that which did not happen? The Soviets already had a small nuclear weapons research program before Hiroshima and Nagasaki, and had been spying on the Manhattan Project. They knew these things were possible, they knew the US had developed and even tested such weapons. Even if the US hadn't used them against Japan, would they not have felt the desire to pursue them anyway? It seems likely. Perhaps the urgency of it might have shifted. But it is quite a jump to assume that the Soviets would have foresworn such a weapon indefinitely.

OK, what if we change it to, "what if the Manhattan Project had never happened?" E.g., what if nuclear weapons had not been pursued in earnest during World War II? The possibility of nuclear weapons was known to scientists globally since 1939. It is true, I think, that the Manhattan Project greatly accelerated the process of their getting built from what it might have otherwise been: no other nation had anything like a real nuclear weapons production program during WWII (the UK, Germany, USSR, and Japan all had very small research programs, not production programs). Certainly the Manhattan Project showed both that it could be done, and gave instructions on several ways it could be done (e.g., the fact that uranium enrichment was possible at scale, the fact that plutonium could be produced in reactors, the fact that you could use both uranium and plutonium as weapons fuel — this was all stuff that was imagined to be the case by many scientists globally, but having it demonstrated, and knowing which methods were successful, is a big difference).

Obviously we cannot really even begin to guess with any accuracy as how world history would have proceeded if the Manhattan Project had never come together. Which is more possible than most people realize — there was nothing "inevitable" about the US government deciding to do this, and the circumstances in which they decided to build the bomb are quite odd and unusual, and involve numerous "small" events (as opposed to "big," structural forces) that could have easily gone other ways. It was, as we say, highly contingent, which is historian-speak for "it didn't have to be the way it did." (Some stuff is less contingent than others; big structural shifts probably happen no matter what. Small choices made by individuals, however, don't seem like they have to. The early Manhattan Project was a lot of small choices, before it built up momentum.)

Given the way the history of science and technology has typically worked, I think one could expect that nuclear weapons would be invented eventually anyway, but at a slower pace. Indeed, one way to think about the Manhattan Project was that it condensed what would have likely been a decades' worth of research and development at a more normal pace into about 2.5 years. That is what made it a "crash" program. There is no doubt that without the Manhattan Project, scientists in many countries would have, after the war, experimented with building nuclear reactors (they were already starting such experiments before the war began in earnest), and upon doing such a thing they would have learned quite a lot about the possibilities of weapons. It is hard to imagine some country not deciding to pursue this vigorously once that seemed like a real possibility. Is it the case that once one country does such a thing, others would follow suit? It seems reasonable to say so, in the absence of any sort of international agreement not to build such weapons.

Could there have been such an international agreement? There is an interesting argument, put forward by some of the scientists who worked on the bomb, that the only way to have possibly any chance at convincing the world to ban nuclear weapons was actually using them on cities, so people could see what they really were. And even that did not really accomplish that aim, despite some real attempts in the late 1940s to do such a thing ("international control of atomic energy").

Anyway, I just say the above to give you an illustration of different ways you can run through the possibilities. It is impossible to know how it would have played out. Perhaps an arms race was inevitable, although it might have been down-shifted in time, which would alter things in ways that are hard to appreciate (e.g., who knows what US or Soviet postwar policy would have been without the idea of nuclear weapons being "out there").