How did scientists determine how many protons, electrons and neutrons are in the atoms of the elements?

An atomic bomb is different from an H-bomb. H-bombs are more violent per equal m. Yet, your question is about how did they determine the particles of the atoms? One way to investigate Hydrogen would be to place the gas in a strong magnetic field and separate the electrons from the nucleons. Then, weigh the two. It's crude enough to get you a good ballpark figure. As for Uranium, Avogadro's number may have been a valuable tool.