Who does the theme music for the Austin Powers movies?

Heaps of people.
https://www.imdb.com/title/tt0118655/soundtrack/

https://www.imdb.com/title/tt0145660/soundtrack/

George S Clinton worked on both GoldMember and The Spy who Shagged Me.

Original Music for The Spy who Shagged Me
George S. Clinton
John Flansburgh (theme song 'Dr. Evil') (as They Might Be Giants)
Susanna Hoffs (song 'Get the Girl')
Susanna Hoffs (song 'Psychedelic Scene Breaks')
John Linnell (theme song 'Dr. Evil') (as They Might Be Giants)
Madonna (song 'Beautiful Stranger')
Debbi Peterson (song 'Get the Girl')
Victoria Peterson (II) (song 'Get the Girl')

Original Music for GoldMember
Greg Camp (song 'Ain't No Mystery')
George S. Clinton
Bruno Coon (song)
Paul De Lisle (song 'Ain't No Mystery')
Steve Harwell (song 'Ain't No Mystery')
Chad Hugo (songs 'Work It Out, 'Boyz')
Mike Myers (song)
Paul Oakenfold (song '1975')
Pharrell Williams (songs 'Work It Out', 'Boys')

Quincy Jones composed the theme 'Soul Bossa Nova', originally for the Canadian TV game show 'Definition', from 1975. Mike Myers, who created the Powers character and being Canadian, made the music the Powers theme because of the above reason.