Green’s Dictionary of Slang

a pair of idling, loafing individuals.

1900		Bulletin (Sydney) 16 June 23/2: A drove of tramps carrying bundles of rags and such bundles as tramps collect. A regiment of Weary Willies and Tired Tims with ragged beards, and smashed slouch-hats, and tin-cans and empty beef-tins dangling by strings from their brace-buttons. [Ibid.] 18 Aug. 14: [cartoon caption, of two vagrants on a park bench] Weary William and Tired Thomas.
1911		Eve. Teleg. 4 July 2/6: A pair of tramps, ideal specimens of the Weary Willie and Tired Tim variety [...] slept without permission in a shed belonging to Mr Thomas Hynd.
1923		Aberdeen Jrnl 11 Dec. 5/6: We have all known that ‘tired feeling’ [...] one that proves so blessed an excuse to the ‘Tired Tim’s’ and ‘Weary Willie’s’ of the community.
1926		Western Dly Press 12 Mar. 9/4: Some of us can remember the ‘Weary Willies’ who spent the best part of their time propping up the pub [...] Now they are concentrated mostly in one place, the Labour Exchange.
1930		Eve. Teleg. (Dundee) 2 Jan. 3/7: ‘Work for all’ is a fine slogan, but Weary Willie and Tired Tim shiver even in a fine day when they read it.