I think you can do it either way, trinity or ABRSM

Music Performance Diploma syllabus from 2005 : ABRSM.org - Exams

they also do jazz exams to grade 5 which you can use instead of your grade 5 theory to then go onto 6,7 or 8.

for your jazz I would buy this

Jamey Aebersold volume 54: Maiden Voyage - Jamey Aebersold Play Alongs - How To Play Jazz

or any of his books, there really good! basically have a 'head'- main melody then an improvised section with all the scales written in the improvised sections

I would also recommend patterns for jazz
Jerry Coker, James Casale, Gary Campbell: Patterns For Jazz (treble clef) - Improvisation - How To Play Jazz

which is all chord/scale work

listen to some jazz tpt players

find a jazz band