1888, Australian, New Zealand, and South African slang for "English immigrant," short for lime-juicer (1857), a nickname given in derisive reference to the British Navy's policy (begun 1795) of issuing lime (n.2) juice on ships to prevent scurvy among sailors. U.S. use is attested from 1918, originally "British sailor, British warship;" extended to "any Englishman" by 1924.

Midway Signs Limey Prof to Dope Yank Talk ["Chicago Tribune" headline, Oct. 18, 1924, in reference to the hiring of William A. Craigie by University of Chicago to begin editing what would become the "Dictionary of American English"]

"British sailor; English person," 1857; see limey. In reference to lime-juice "the juice of the lime" (1704), which was popular 19c. as an antiscorbutic and stocked on vessels bound on long voyages. Lime-water (1670s) was the usual word for "solution of lime (n.1) in water."