Fashionable greeting in Conservative clubs (4)
I believe the answer is:
chic
'fashionable' is the definition.
(chic means stylish or fashionable)
'greeting in conservative clubs' is the wordplay.
'greeting' becomes 'hi' (hi is a kind of greeting).
'in' is an insertion indicator.
'conservative' becomes 'C' (abbreviation commonly used in UK politics).
'clubs' becomes 'c' (abbreviation used in card games such as bridge).
'c'+'c'='cc'
'hi' placed into 'cc' is 'CHIC'.
(Other definitions for chic that I've seen before include "style?" , "Elegantly stylish" , "Fashionable good taste" , "Fashionably smart" , "Modish" .)