In the past, not quite everyone ready to respond (2,4)
I believe the answer is:
on call
'ready to respond' is the definition.
(I've seen this before)
'in the past not quite everyone' is the wordplay.
'in the past' becomes 'once' (once upon a time).
'not quite' means to remove the last letter (most but not all of the word).
'everyone' becomes 'all' (all people).
'once' with its final letter removed is 'onc'.
'onc'+'all'='ON CALL'
(Other definitions for on call that I've seen before include "available in emergencies?" , "Like a doctor, perhaps" , "Available if needed" , "ready to respond" , "Contactable" .)