Dimwit mostly with nothing to say about Oxbridge degree (6)
I believe the answer is:
muppet
'dimwit' is the definition.
(I know that dimwit can be written as muppet)
'mostly with nothing to say about oxbridge degree' is the wordplay.
'mostly' means to remove the last letter (most but not all of the word).
'with nothing to say' becomes 'mute' (unable to speak).
'about' means one lot of letters goes inside another.
'oxbridge degree' becomes 'PPE' (Politics, Philosophy and Economics degree).
'mute' with its final letter taken away is 'mut'.
'mut' going around 'ppe' is 'MUPPET'.
(Other definitions for muppet that I've seen before include "Idiot" , "Kermit or Miss Piggy, say" , "Weak person (informal)" , "Soft - hearted TV character" , "Fool - Jim Henson creation" .)