Not just any flash British gangsters (3,3)
I believe the answer is:
the mob
'gangsters' is the definition.
(I've seen this before)
'not just any flash british' is the wordplay.
'not just any' becomes 'the' (I've seen this in another clue).
'flash' becomes 'mo' (both mean a very short period of time).
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'the'+'mo'+'b'='THE MOB'
(Other definitions for the mob that I've seen before include "common people" , "Organised criminals" , "gangsters" , "lot to which 20 [JAMES GANDOLFINI] in character belonged" .)