Conservative prig, when out of head, is vulgar (5)
I believe the answer is:
crude
'vulgar' is the definition.
('crude' can be a synonym of 'vulgar')
'conservative prig when out of head' is the wordplay.
'conservative' becomes 'c' (abbreviation commonly used in UK politics).
'prig' becomes 'prude' (I've seen this before).
'when out of head' means to remove the first letter.
'prude' with its initial letter taken away is 'rude'.
'c'+'rude'='CRUDE'
'is' is the link.
(Other definitions for crude that I've seen before include "Coarse and unrefined" , "Not refined, oil say" , "Rude; raw" , "Petroleum" , "Unrefined; makeshift" .)