Conservative speaker losing head, grabbing university steward (7)
I believe the answer is:
curator
'steward' is the definition.
(curator is a kind of steward)
'conservative speaker losing head' is the wordplay.
'conservative' becomes 'cu' (I am not sure about this - if you are sure you should believe this answer much more).
'speaker' becomes 'orator' (orator is a kind of speaker).
'losing head' means to remove the first letter.
'orator' with its initial letter taken away is 'rator'.
'cu'+'rator'='CURATOR'
'grabbing university' acts as a link.
This may not be right. Some or all of it may belong to another bit of the clue.
(Other definitions for curator that I've seen before include "Artistic director" , "Museum official" , "Museum custodian" , "Guardian" , "Person in charge of a museum" .)