Green’s Dictionary of Slang

(orig. US) the female breasts.

1909		L.W. Payne Jr ‘Word-List From East Alabama’ in DN III:v 352: ninny, n. Milk from the breast; also one of the breasts.
1949		Monteleone Criminal Sl. (rev. edn).
1969		E. Leonard Moonshine War 76: The girl was certainly pretty, a thin little thing but with a pair of grown woman’s ninnies.
1979		Frank Zappa ‘Fembot in a Wet T-Shirt’ 🎵 When the water gets on ’em / Their ninnies get rigid / ’N look pretty bold.