Budding musician finally on the way up (7)
I believe the answer is:
nascent
'budding' is the definition.
(I've seen this before)
'musician finally on the way up' is the wordplay.
'finally' suggests the final letters.
'on' is a charade indicator (letters next to each other).
'the way up' becomes 'ascent' (I've seen this before).
The last letter of 'musician' is 'n'.
'n'+'ascent'='NASCENT'
(Other definitions for nascent that I've seen before include "Sent can (anag.)" , "Just beginning existence" , "Emerging, in the process of being born" , "starting to mature" , "Freshly generated" .)