PM to resign? Has to go about it tortuously (7)
I believe the answer is:
asquith
'pm' is the definition.
(I've seen this before)
'go about it tortuously' is the wordplay.
'go' becomes 'quit' (I've seen this before).
'about it' indicates putting letters inside.
'tortuously' becomes 'ash' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'quit' put within 'ash' is 'ASQUITH'.
'to resign? has to' acts as a link.
This may not be right. Some or all of it may be part of another bit of the clue.
(Other definitions for asquith that I've seen before include "Former Prime Minister" , "Herbert - - , WW1 PM" , "Liberal prime minister, 1908 to 1916" , "Liberal statesman" , "Old PM" .)