> Had information on warheads and the like been properly contained, maybe other countries would not have so easily developed the bomb.

The Soviets had people inside the Manhattan Project / Los Alamos. As the US made progress that information was fed to the Soviets/Russians.

* https://en.wikipedia.org/wiki/Klaus_Fuchs

* https://en.wikipedia.org/wiki/Atomic_spies

Not so easily, but they would have done it. Once it was known, there wouldn't have been any way to stop Stalin. His paranoia knew no limits. And then there would have been dozens, hundreds when the war would break out, nobody would be scared to use them.

Unfortunately, democracies have their downsides. One of them is that "properly containing" things is extremely hard. Soviet spy network was vast and infiltrated all tiers of US society, and without instituting some extremely draconian policies, compared to which McCarthy would look like a hippie, I don't see how it would be possible to prevent it. Especially given a lot of top scientists were, unfortunately, if not communists themselves than quite friendly with communists and would probably refuse to cooperate with any draconian regime that would be capable of suppressing the communist networks.