Does its type change based on the Ability or does it just stay the Pokémon's Hidden Power type?
Well no because though Hidden Power appears as a Normal type attack it's not. It cannot be a normal type move therefore it stays the type it was given.
Source: Knowledge + I looked this up
Bulbapedia page on Hidden Power (if you wanted one of my sources)
It stays the Pokemon's hidden power type, as shown in this replay (aeriliate works the same way)