Shortly after niece or nephew starts (4)
I believe the answer is:
anon
'shortly' is the definition.
(I've seen this before)
'after niece or nephew starts' is the wordplay.
'starts' indicates taking the first letters.
The first letters of 'after niece or nephew' is 'ANON'.
(Other definitions for anon that I've seen before include "Well-published writer" , "prolific author" , "Presently - or unknown" , "Soon - nobody knows me" , "Soon or presently" .)