Green’s Dictionary of Slang

1. the penis.

1960		J. Franklyn Dict. of Rhy. Sl.
1972		Dodson & Saczek Dict. of Cockney Rhy. Sl.
1978		Maledicta II:1+2 (Summer/Winter) 117: More examples are Pat and Mick = ‘prick’, Rory O’More = ‘whore’ (also scrubber, slag, etc.), and Alphonse = ponce (‘pimp’).

2. (Aus.) a pick.

1941

Aus. Women’s Wkly 26 July 22/2: The rhyming slang of the Cockneys still has its adherents [...] ‘Pat and Mick’ for pick.