My answer is yes, with low-moderate certainty. I still think the USA would have developed it first, and I think this is what is suggested to us by the GDP trends of the US versus basically everywhere else post-WW2.
Take this all with more than a few grains of salt. I am by no means an expert in this territory. But I don't shy away from thinking about something just because I start out sounding like an idiot. Also take into account this is post-hoc, and 1940 Manhattan Project me would obviously have had much, much less information to work with about how things actually panned out. My answer to this question should be seen as separate to the question of whether I think dodging the Manhattan Project would have been a good bet, so to speak.
Most historians agree that Japan was going to lose one way or another by that point in the war. Truman argued that dropping the bomb killed fewer people in Japan than continuing, which I agree with, but that's a relatively small factor in the calculation.
The much bigger factor is that the success of the Manhattan Project as an ultimate existence proof for the possibility of such weaponry almost certainly galvanized the Soviet Union to get on the path of building it themselves much more aggressively. A Cold War where one side takes substantially longer to get to nukes is mostly an obvious x-risk win. Counterfactual worlds can never be seen with certainty, but it wouldn't surprise me if the mere existence proof led the USSR to actually create their own atomic weapons a decade faster than they would have otherwise, by e.g. motivating Stalin to actually care about what all those eggheads were up to (much to the terror of said eggheads).
This is a bad argument to advance when we're arguing about e.g. the invention of calculus, which as you'll recall was coinvented in at least 2 places (Newton with fluxions, Liebniz with infinitesimals I think), but calculus was the kind of thing that could be invented by one smart guy in his home office. It's a much more believable one when the only actors who could have made it were huge state-sponsored laboratories in the US and the USSR.
If you buy that, that's 5 to 10 extra years the US would have had in order to do something like the Manhattan Project, but in much more controlled, peace-time environments. The atmosphere-ignition prior would have been stamped out pretty quickly by later calculations of physicists to the contrary, and after that research would have gotten back to full steam ahead. I think the counterfactual US would have gotten onto the atom bomb in the early 1950s at the absolute latest with the talent they had in an MP-less world. Just with much greater safety protocols, and without the Russians learning of it in such blatant fashion. Our abilities to detect such weapons being developed elsewhere would likely have also stayed far ahead of the Russians. You could easily imagine a situation where the Russians finally create a weapon in 1960 that was almost as powerful as what we had cooked up by 1950.
Then you're more or less back to an old-fashioned deterrence model, with the twist that the Russians don't actually know exactly how powerful the weapons the US has developed are. This is an absolute good: You can always choose to reveal just a lower bound of how powerful your side is, if you think you need to, or you can choose to remain totally cloaked in darkness. If you buy the narrative that the US were "the good guys" (I do!) and wouldn't risk armaggedon just because they had the upper hand, then this seems like it can only make the future arc of the (already shorter) Cold War all the safer.
I am assuming Gorbachev or someone still called this whole circus off around the late 80s-early 90s. Gotta trim the butterfly effect somewhere.