Military unit’s chaplain quietly becoming Conservative (5)
I believe the answer is:
cadre
'military unit's' is the definition.
(I've seen this before)
I can't explain the rest of the clue.
(Other definitions for cadre that I've seen before include "Core group -- cared (anag)" , "revolutionary cell" , "Nucleus, framework" , "activist group" , "Group of revolutionaries" .)