Green’s Dictionary of Slang

alone; thus in phr. on one’s Darby, by oneself.

1942		G. Kersh Nine Lives Bill Nelson 31: I say a guy’ll die as he was born—on his Derby-and-Joan.
1957		G. Kersh Fowlers End (2001) 187: Provided you don’t live like a pig—in your Darby-and-Joan, for yourself.
1960		J. Franklyn Dict. of Rhy. Sl.
1970		C. Brown Down All the Days 88: You’re bloody Darby-and-Joan now, me bucko, thinking the sun shines out of her, but wait till you’ve a gang of snotty-nosed, shitty-arsed crying little demons around you.