Quiet road going east away from British city (6)
I believe the answer is:
mumbai
'city' is the definition.
(Indian city)
'quiet road going east away from british' is the wordplay.
'quiet' becomes 'mum' (I've seen this before).
'road' becomes 'ai' (I've seen this in other clues).
'going east away from' is a charade indicator (letters next to each other) (some letters go to the right or east of others).
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'ai' put after 'b' is 'bai'.
'mum'+'bai'='MUMBAI'
(Other definitions for mumbai that I've seen before include "New name for Bombay" , "It used to be called Bombay" , "Formerly Bombay" , "City once called Bombay" , "Modern name for Bombay" .)