According the Merriam-Webster's great Collegiate Dictionary, the earliest known use of galaxy comes from Chaucer sometime before 1385 in a poem called The House of Fame:
Se yonder, loo, the Galaxie,
Which men clepeth the Milky Wey,
For hit is whit.
(See yonder, lo, the galaxy
Which men call the Milky Way
For it is white.)
Clearly, then, the name Milky Way was common at least in England at this time. Galaxy wasn't used as a generic term for other massive star systems until the 19th century.