Tempo managed to greet everyone first (10)
I believe the answer is:
allegretto
'tempo' is the definition.
(allegretto is a kind of tempo)
'managed to greet everyone first' is the wordplay.
'managed' is an anagram indicator.
'everyone' becomes 'all' (all people).
'first' means one lot of letters go next to another.
'to'+'greet'='togreet'
'togreet' anagrammed gives 'egretto'.
'egretto' after 'all' is 'ALLEGRETTO'.
(Other definitions for allegretto that I've seen before include "with reasonable rapidity" , "at a fairly brisk rate" , "quite quickly" , "Moving fairly briskly (music)" , "somewhat quick" .)