Give away secret British science establishment (4)
I believe the answer is:
blab
'give away secret' is the definition.
(to blab is to reveal a secret)
'british science establishment' is the wordplay.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'science establishment' becomes 'lab' (laboratory).
'b'+'lab'='BLAB'
(Other definitions for blab that I've seen before include "Spill the beans" , "Tell tales" , "Talk indiscreetly" , "Give the game away" , "Let the cat out of the bag" .)