Green’s Dictionary of Slang

an unspecified part of the body, presumably the nose; thus usu. in the phr. a punch up the bracket.

1959		Galton & Simpson ‘Hancock’s Half-Hour’ ser. 5 [radio ascript] Are you asking for a punch up the bracket?
1984		Partridge DSUE (8th edn) 128/1: bracket A vague, unspecified part of the body, prob. the nose; it occurs in threat [...] popularised by the BBC radio comedy programme ‘Hancock’s Half-hour’ of the 1950s.
1996		S. Maloney Brush-Off (1998) 68: He had a bracket like a Borgia pontiff in a Titian portrait.
2006		www.laughingpoliceman.com 🌐 Imagine you are out shopping, or in the pub and you inadvertently call someone a CUNT. What happens……….nothing! (or possibly a punch up the bracket if he is bigger than you!).