British PM's remains left inside (7)

I believe the answer is:

asquith

I'm a little stuck... Click here to teach me more about this clue!

'british pm's' is the definition.
'Asquith' can be an answer for 'pm's' (H. H. Asquith, UK prime minister 1908-1916). I am not certain of the 'british' bit.

'remains left inside' is the wordplay.
'remains' becomes 'ash' (remains of a fire).
'left' becomes 'quit' (I've seen this in another clue).
'inside' means one lot of letters goes inside another.
'ash' placed around 'quit' is 'ASQUITH'.

Can you help me to learn more?

(Other definitions for asquith that I've seen before include "Ex-PM" , "Former PM" , "Liberal statesman" , "H H ---, Liberal PM" , "Former Prime Minister" .)

I've seen this clue in The Telegraph.

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