Very nearly everybody had an objection (3,3)
I believe the answer is:
all but
'very nearly' is the definition.
('all but' can be a synonym of 'very nearly')
'everybody had an objection' is the wordplay.
'everybody' becomes 'all' (I've seen this before).
'had' means one lot of letters go next to another.
'an objection' becomes 'but'.
'all'+'but'='ALL BUT'
(Other definitions for all but that I've seen before include "'Just short of (3,3)'" , "Not quite" , "As good as" , "Very little short of, almost there" , "Very little short of, nearly" .)