Premature but nearly doesn't start (5)
I believe the answer is:
early
'premature' is the definition.
(similar in meaning)
'nearly doesn't start' is the wordplay.
'doesn't start' means to remove the first letter.
'nearly' with its initial letter taken off is 'EARLY'.
'but' acts as a link.
(Other definitions for early that I've seen before include "Infancy, when Wordsworth had 5 across of 5 down" , "near beginning" , "Before the usual time" , "in the morning?" , "Immature" .)