"automatic coin-operated public laundry," 1946, originally (1942) a proprietary name by Westinghouse for a type of automatic washing machine; from laundry + ending probably suggested by automat. Also used for public clothes-washing places in U.S. were washateria (1935), laundrette (1945). The Westinghouse machine was popular after the war and was available with coin chutes and timers.
[laundromat etymology, laundromat origin, 英语词源]
Old English wæscing "action of washing clothes," verbal noun from wash (v.). Meaning "clothes washed at one time" is from 1854. Washing machine attested from 1754.