Celebrate? It might be deflating for a Cockney (3,4,4,4)
I believe the answer is:
let ones hair down
'celebrate?' is the definition.
(to let one's hair down is to party)
'it might be deflating for a cockney' is the wordplay.
I cannot really understand how this works, but
'it' could be 't' (abbreviation. e.g. in 'tis) and 't' is present in the answer.
'a' could be 'one' and 'one' is located in the answer.
This may be the basis of clue (or it may be nonsense).
(Other definitions for let ones hair down that I've seen before include "have a lot of fun" , "act without reserve" , "Make whoopee" , "lowering the shock level" , "Be a disgrace to Barnet" .)