European punching body part — nose gets swollen (9)

I believe the answer is:

tumescent

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'swollen' is the definition.
(synonyms)

'european punching body part nose' is the wordplay.
I cannot really understand how this works, but
'european' could be 'e' (abbreviation e.g. EU) and 'e' is located in the answer.
'part' could be 'tum' (tum is a kind of part) and 'tum' is found within the answer.
'nose' could be 'scent' (synonyms) and 'scent' is found within the answer.
No letters remain.
This explanation may well be incorrect...

'gets' acts as a link.

Can you help me to learn more?

(Another definition for tumescent that I've seen is " swollen".)

I've seen this clue in The Guardian.

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