Probably Not Much Earlier Than We Did
The problem is less with the concept of a nuclear bomb - Enrico Fermi famously speculated as to the liberation of atomic energy for destructive purposes in the early 20s - and much, much more with the necessary materials science required to make the fissile material. It took thousands of centrifuges, billions of dollars, the accidental discovery of Teflon...
The "billions of dollars" part is also a big deal. It was \$2B to get to the first bomb's worth of material, in 1942 dollars. That would only have been $1.2B in 1900, assuming that individual tasks wouldn't cost more in real values (which it would), but that would have been 10% of the entire GNP of the time. It's hard to imagine, barring an existential threat (which WWI was not!) a country spending an amount of money nearly impossible to contemplate on something that atomic theory didn't yet support.
You'd also need advances in mining. Chemical leaching in place of traditional smelting.
By the time the US felt it needed an atomic bomb, it already had plenty of scientists who postulated that it was possible... and had all the other tools it needed, as long as it had the motivation to spend an enormous sum of money. In 1900, it didn't have the technology, nor any of the other tools. Knowing that it was possible wouldn't magically advance those - particularly lacking the motivation.
Per @AlexP's comment, the centrifuges weren't even successful, though they would be eventually. But I should also point out that prior to WWII, there wasn't really a country on earth that would be called a "superpower". Prior to WWI, the United States barely had an army!