Progress after ousting Conservative member (4)
I believe the answer is:
limb
'member' is the definition.
(member can mean a limb)
'progress after ousting conservative' is the wordplay.
'progress' becomes 'climb' (climb can mean to progress).
'after ousting' is a deletion indicator.
'conservative' becomes 'C' (UK political abbreviation).
'climb' with 'c' taken out is 'LIMB'.
(Other definitions for limb that I've seen before include "Bough" , "Jointed appendage" , "Branch - arm - leg" , "Branch of a tree (risky to go out on)" , "Arm, leg or wing" .)