Green’s Dictionary of Slang

tit willow n.

a pillow.

1932		‘P.P.’ Rhy. Sl. 11: Stick yer ‘lump of lead’ on the ‘tit-willow’ and ’ave a bo-peep, ’cos yer ‘elephants.’.
1960		J. Franklyn Dict. of Rhy. Sl. 93/2: I woke up wiv a marf like a carsey an’ I couldn’t lift me lump-o’-lead orf da titwillow to git at the fisherman’s.
1979		R. Barker Fletcher’s Book of Rhy. Sl. 39: A man without a tit willow to lay his head on.
2002		B. Kirkpatrick Wicked Cockney Rhy. Sl.