The OED on-line says:

2. slang. An infantile name for the penis. Also Comb., as willy-warmer.
1905 Eng. Dial. Dict. Suppl. 178/2 Willy, the male organ; a slang name for a child's penis. Cum., Wm. 1972 Listener 22 June 841/3 The gallant soldier-boys are afflicted with 'syph, darling' ('their willies rot away'). 1975 Observer 7 Dec. 27/3 Joky gifts are speechlessly embarrassing; this season's dud is a woolly willy-warmer. 1977 J. WILSON Making Hate ix. 113 A younger male [baboon]..fingered its crimson penis... 'It's playing with its willie!' Nicky squealed. 1985 P. ANGADI Governess x. 93 We used to hold each other's willies... We didn't know about sex then.

So it seems to be a relatively recent usage.