Bank worker to inform the cockney girl (6)
I believe the answer is:
teller
'bank worker' is the definition.
(I've seen this in another clue)
'inform the cockney girl' is the wordplay.
'inform' becomes 'tell' (telling is a kind of informing).
'the cockney girl' becomes 'er' ('her' - 'her' refers to a girl or woman - said with a cockney accent, dropping the 'H').
'tell'+'er'='TELLER'
'to' is the link.
(Other definitions for teller that I've seen before include "Bank official or reporter" , "one gives relation" , "who worked on the hydrogen bomb" , "Election official" , "One who instructs" .)