First to arrive in Leeds lost little time (6)
I believe the answer is:
eldest
'first to arrive' is the definition.
(I've seen this in another clue)
'leeds lost little time' is the wordplay.
'lost' is an anagram indicator.
'little time' becomes 't' (abbreviation for time).
'leeds' is an anagram of 'eldes'.
'eldes'+'t'='ELDEST'
'in' is the link.
(Other definitions for eldest that I've seen before include "Of greatest age" , "Most senior of a group" , "Most senior of two" , "Such as Cain, say" , "most early in origin" .)