Since a hydrogen atom is basically just a proton with an electron, and a hydrogen nucleus IS a proton, it's difficult to say exactly who discovered it because doing so gets caught up with asking who discovered the element and atomic nature of hydrogen. The person I'd give credit, though, would be Ernest Rutherford, who discovered that atoms of elements other than hydrogen contained hydrogen nuclei, which implies that whatever hydrogen is, it's one of the building blocks of the other elements as well as itself.

I believe it was Ernest Rutherford who kind of discovered the proton. He did many experiments on radioactivity and found that there was a positively charged thing, the proton. He named it from the Greek work "protos" which means "first".