Green’s Dictionary of Slang

woodie n.1

also wood, woody

a Wills Woodbine cigarette.

[UK]M. Harrison Reported Safe Arrival 65: ’Ere, giss a light fer me Wood!
[UK](con. WWII) B. Aldiss Soldier Erect 123: I was smoking before I was weaned [...] filching Woodies off my elder brothers.
[UK]M. Simpson ‘Prufrock Scoused’ Catching Up with Hist. 22: The owld geezers on ther tod / smokin ther Woodies.
(con. 1953) F. Butler Up the Snicket 15: The Woodies were shared out (‘every cigarette bearing the signature W.D. & H.O. Wills’).
[UK]H. Mantel Beyond Black 204: ‘They don’t make Woodbine no more.’ ‘What, they don’t make Woodies? Why not?’.