One's old man going round demanding immediate attention (7)
I believe the answer is:
exigent
'demanding immediate attention' is the definition.
(I've seen this before)
'one's old man going round' is the wordplay.
'one' becomes 'i' (Roman numeral).
'old' becomes 'ex' (prefix meaning former or onetime).
'man' becomes 'gent' (gent is a kind of man).
'going round' means one lot of letters goes inside another (some letters go around others).
'ex'+'gent'='exgent'
'i' placed inside 'exgent' is 'EXIGENT'.
(Other definitions for exigent that I've seen before include "Pressing and urgent" , "Exacting, requiring much" , "Urgent and exacting" , "Pressing; demanding" , "Demanding immediate attention" .)