Was there any conspiracy surrounding the Hiroshima/Nagasaki bombings before they took place, meaning the conspiracists were proved correct?

Yes and no.

There were leaks and suspicions. There were lots of rumors. There were people, in the US and abroad, who more or less figured out that the US must be making a bomb, and amongst journalists in DC there was an active rumor mill that there was a project called "Manhattan" that was going to do something big soon.

There were people who noticed the brain drain, for sure. (The Soviets did, among others. So did Indian scientists.) But it should be noted that this brain drain was in the context of a massive brain drain during World War II. It wasn't specific to the atomic bomb — all scientists in all disciplines got involved in war projects. That wasn't a secret. The specifics were.

TIME magazine wrote in May 1942:

Exploration of the atom — chief interest of physicists — has come to a stop. ... Such facts as these add up to the biggest scientific news of 1942: that there is less and less scientific news. Technical journals are thinner by as much as 50%, and they will get more so: much of the research now published was completed a year ago before the conversion of U.S. science to wartime uses had reached all-out proportions. A year ago one out of four physicists was working on military problems; today, nearly three out of four. And while news from the world’s battlefronts is often withheld for days or weeks, today’s momentous scientific achievements will not be disclosed until the war’s end. ... Pure research is not secret now. In most sciences it no longer exists.

There are lots of other things along these lines. Sometimes the stories mentioned atomic bombs as a possibility (nuclear fission was widely known after 1939), but in the same stories they also mentioned things like freeze bombs and rays that could stop airplane engines and death rays and things like that. Nothing very specific. Just a general interest in fantastical war technologies, spun off in part because of the excitement over the German's V-1 and V-2 rockets.

The closest things to real "conspiracy theories" were a few people who wrote to their congressmen alleging that the various project sites (e.g. Hanford or Oak Ridge, which were conspicuously large and secret to those who lived in the areas) were really just massive government mismanagement projects. My favorite allegation was that Oak Ridge was a Communist colony developed by Eleanor Roosevelt, a model that would later become a template for the whole country. None of those people had any idea what was going on, though.

I wrote about the worst of the many Manhattan Project leaks here, which is the closest anyone got to anything like a conspiracy theory. As you can see, it is not much of a theory.

I don't really consider this sort of thing to fall into the category of "conspiracy theorizing." There was conspiracy theorizing during World War II and after, but it was centered around questions about Roosevelt knowing about Pearl Harbor, Communist subversion, things like that. It was not generally about new weapons.

For someone looking for the Manhattan Project, it was not hard to find. There were signs everywhere. It was just too big to effectively hide. It employed some 600,000 people — nearly 1% of the entire US civilian labor force during World War II. About one out of 250 Americans alive were working on the bomb.

But if you weren't looking, you wouldn't see it, even if parts of it were right in front of your face. The Germans and the Japanese apparently didn't look for it very closely. The Soviets did, and found it. They gave it a very appropriate code-name: "ENORMOZ."