Green’s Dictionary of Slang

1. to annoy someone, to infuriate someone.

1909		T.A. Dorgan in Zwilling TAD Lex. (1993) 89: Fedink’s Speed Got Bunk’s Nanny Something Awful.
1914		R. Lardner ‘Horseshoes’ in Coll. Short Stories (1941) 256: ‘Good night, Horseshoes,’ he says. That got my nanny this time. ‘Shut up, you lucky stiff!’ I says.
1926		Wood & Goddard Dict. Amer. Sl.
1930		J. Tully Shadows of Men 247: He’s jist tryin’ to git your nanny.
1942		Davis & Wolsey Call House Madam (1943) 256: She could get red-hot and strike with the speed of a scorpion if somebody got her nanny.

2. (Aus.) to play around, to act foolishly and irresponsibly.

1993

(ref. to 1940s-50s) Aus. Word Map 🌐 shaiack the nanny. To act in a foolish and irresponsible manner; to ‘horse around’: Don't shaiack the nanny! [...] This was used in this area and parts of Victoria around the 1940s and perhaps into the mid 1950s.