Green’s Dictionary of Slang

1. a chat, a conversation.

1959		F. Norman Fings I i: ’E’s ’avin’ a right old bunny with Fred.
1979		F. Norman Too Many Crooks Spoil the Caper 16: Maybe he even thought you’d had a little bunny with the bogies yourself.
1989		in G. Tremlett Little Legs 75: We’d have a little bunny about this and that.

2. a talkative person.

1984		Partridge DSUE (8th edn) 158/2: since ca. 1954.
2108		(con. 1930s) M. McGrath Pie & Mash 57: Marion is [...] a good talker. A ‘right old bunny rabbit’.