Uncaring, cold as usual, snubbing our lot (6)
I believe the answer is:
casual
'uncaring' is the definition.
(being casual can mean not caring about something)
'cold as usual snubbing our lot' is the wordplay.
'cold' becomes 'c' (eg on taps).
'snubbing' indicates named letters should be taken away.
'our lot' becomes 'us'.
'usual' with 'us' removed is 'ual'.
'c'+'as'+'ual'='CASUAL'
(Other definitions for casual that I've seen before include "Informal or not showing commitment" , "Informal, dress say" , "Irregular" , "Without formality" , "Relaxed, unconcerned" .)