Public outside nearly until morning? (9)
I believe the answer is:
overnight
'until morning?' is the definition.
(through the night)
'public outside nearly' is the wordplay.
'public' becomes 'overt' (both can mean well-known or unconcealed).
'outside' means one lot of letters goes inside another.
'nearly' becomes 'nigh' (synonyms).
'overt' placed around 'nigh' is 'OVERNIGHT'.
(Other definitions for overnight that I've seen before include "Extending from dusk to dawn" , "Lightning" , "from one day to the next" , "Until morning; suddenly" , "Lasting from dusk to dawn" .)