Green’s Dictionary of Slang

dubber n.²

the mouth.

1845		G.W.M. Reynolds Mysteries of London vol. 2 142: Mum your dubber. Keep your mouth shut.
1860		Hotten Dict. of Modern Sl. etc. (2nd edn) 134: DUBBER, the mouth; ‘mum your dubber,’ hold your tongue.
1873		Sl. Dict.

In phrases

dubber-mum(m)ed (adj.) [mum adj.]

silent.

1789		G. Parker Life’s Painter 136: Aye, do, why should you be dubber-mum’d? there’s no hornies, traps, scouts, nor beak-runners amongst them.
1833		‘Rowling Joey & Moll Blabbermums’ in Corinthian in Spedding & Watt (eds) Bawdy Songbooks (2011) IV 42: [as 1789].