Green’s Dictionary of Slang

Uncle Dick adj.

ill, sick.

1946		St. Vincent Troubridge ‘Some Notes on Rhyming Argot’ in AS XXI:1 Feb. 46: half-a-lick. Sick. (Origin uncertain, but probably American.) Agreed. The English equivalent is Uncle Dick.
1969		S.T. Kendall Up the Frog.
1977		D. Powis Signs of Crime 206: Uncle Dick Same as Tom and Dick.
1980		Barltrop & Wolveridge Muvver Tongue 85: Phrases for being in poor health have not altered [...] ‘a bit dicky’ (probably from ‘Uncle Dick’, sick).