Cuckoo nest he reveals with no adult, still (12)
I believe the answer is:
nevertheless
'still' is the definition.
(I know that still can be written as nevertheless)
'cuckoo nest he reveals with no adult' is the wordplay.
'cuckoo' is an anagram indicator (I've seen this in other clues).
'with no' suggests deleting specific letters.
'adult' becomes 'a' (abbreviation).
'nest'+'he'+'reveals'='nesthereveals'
'nesthereveals' with 'a' removed is 'nestherevels'.
'nestherevels' is an anagram of 'NEVERTHELESS'.
(Other definitions for nevertheless that I've seen before include "Still" , "Notwithstanding" , "In spite of everything" , "anyhow" , "However" .)