Green’s Dictionary of Slang

(UK tramp) methylated spirits (orig. combined with French polish); thus finish-drinker.

1864		Guardian 24 Dec. 4/2: A liquid known as ‘finish,’ a compound of methylated spirits and French polish [has] come into vogue.
[1875		Ure’s Dict. Arts I 58: Methylated spirit can be procured also in small quantities [...] containing in solution 1 oz. to the gallon of shellac, under the name of ‘finish’].
1984		Partridge DSUE (8th edn) 393/1: —1932.