Green’s Dictionary of Slang

milky adj.²

1. cowardly.

1893		‘’Arriet on Labour’ in Punch 26 Aug. 88/2: My young man [...] whose temper’s really milky / Whose ’art is soft as ’is merstarche — and that is simply silky.
1927		(con. 1920s) S. Lewis Elmer Gantry 375: But he was not milky. He was staring hard enough.
1938		G. Greene Brighton Rock (1943) 47: You aren’t milky, are you?
1954		R. Fabian London After Dark 83: Boastful young spivs who will do anything rather than admit to being ‘milky’.
1956		J. Curtis Look Long Upon a Monkey 27: Before you started giving anyone the strength of anything, you always went a bit milky.
1961		(con. 1920s) J. Sparks Burglar to the Nobility 88: I never once [...] heard Speedles cry for mercy or anything milky like that. He chose to laugh instead of screaming.

2. sentimental, ‘soft’.

1961

(con. 1920s) J. Sparks Burglar to the Nobility 67: It probably sounds a bit milky if I was to say now that the reason which makes thieves and villains get through their ill-gotten wages so sharpish is they must inside themselves feel somehow guilty.

In compounds

milky beak (n.) [beak n.¹ (1)]

(UK Und.) a drunken magistrate.

c.1850

Duncombe New and Improved Flash Dict. n.p.: Beak, milky a magistrate who will forego the strict duty of his office for a good treat of wine, &c.