Support British oldster losing daughter (7)
I believe the answer is:
bolster
'support' is the definition.
('bolster' can be a synonym of 'support')
'british oldster losing daughter' is the wordplay.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'losing' indicates named letters should be taken away.
'daughter' becomes 'd'.
'oldster' with 'd' taken out is 'olster'.
'b'+'olster'='BOLSTER'
(Other definitions for bolster that I've seen before include "support for head after retiring" , "Strengthen - long pillow" , "Provide reinforcement" , "Hold up" , "Extended pillow" .)