In fact, "B'way" has been in continuous use since the beginning of the New York Times in the 1850s. I would say that "B'way" was probably used in the 1830s -- the same decade that gave us "O. K."
21 October 1852, New York Daily Times, pg. 8:
...east side of B'way east of B'dway B'way and 9th av. ...