Exact time to accept lawmaker making boast (7)

I believe the answer is:

trumpet

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'boast' is the definition.

'exact time to accept lawmaker' is the wordplay.
'exact' becomes 'true' (I've seen this in another clue).
'time' becomes 't' (abbreviation).
'to accept' indicates putting letters inside.
'lawmaker' becomes 'mp' (I can't justify this - if you can you should believe this answer much more).
'true'+'t'='truet'
'truet' enclosing 'mp' is 'TRUMPET'.

'making' acts as a link.

Can you help me to learn more?

(Other definitions for trumpet that I've seen before include "Loudly sound - instrument" , "Orchestral or military instrument" , "Loud cry" , "Centre of daffodil flower" , "beastly noise" .)

I've seen this clue in The Independent.

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