Army officer and doctor, each ringing when desperate (4-4)
I believe the answer is:
last-gasp
'desperate' is the definition.
(I've seen this before)
'army officer and doctor each ringing when' is the wordplay.
'army officer' becomes 'lt' (abbreviation for lieutenant).
'and' is a charade indicator (letters next to each other).
'doctor each' becomes 'gasp' (I can't explain this - if you can you should give a lot more credence to this answer).
'ringing' means one lot of letters goes inside another.
'when' becomes 'as' (eg 'when I walked' means 'as I walked').
'lt'+'gasp'='ltgasp'
'ltgasp' going around 'as' is 'LAST-GASP'.
(Other definitions for last-gasp that I've seen before include "Done in desperation at the final moment" , "'Final breath (4,4)'" , "Done in desperation (or at the point of death?)" , "Done at the latest possible moment" , "Final exhalation before death" .)