Green’s Dictionary of Slang

a costermonger’s handkerchief, coloured light or dark blue.

1839		W.A. Miles Poverty, Mendicity and Crime; Report 115: The new term for handkerchiefs is a Billy, for which pickpockets have peculiar terms known only in the trade. [...] watersman, sky-coloured, worth 3s.
1847		G.W.M. Reynolds Mysteries of London III 85/1: A Stranger—looked like a spunk fencer. Green king’s-man, water‘s-man, yellow fancy and yellow-man .
1859, 1860		Hotten Dict. of Modern Sl. etc.
1873		Sl. Dict.
1882		Sydney Sl. Dict. (2 edn) 9: Watersman - A sky colored silk handkerchief.