Cutting speed unusually learner driver, American, went by (7)
I believe the answer is:
elapsed
'went by' is the definition.
('elapse' can be a synonym of 'go by')
'cutting speed unusually learner driver american' is the wordplay.
'cutting' is an insertion indicator (some letters cut their way into a word).
'unusually' indicates anagramming the letters.
'learner driver' becomes 'l' (learner drivers have L-plates on their cars).
'american' becomes 'a' (common abbreviation - e.g. in organisation names).
'speed' is an anagram of 'epsed'.
'l'+'a'='la'
'epsed' enclosing 'la' is 'ELAPSED'.
(Other definitions for elapsed that I've seen before include "Pleased (anag.)" , "Went by" , "Pleased to have passed" , "slipped away" , "Passed, of time" .)