In 1959 Bailey was engaged by the Australian theatrical producers J.C. Williamson Limited to play the part of Professor Henry Higgins in their production of the Lerner & Lowe musical My Fair Lady. The production was a duplicate of the New York City production. Although Bailey's was not a name that could attract large audiences in Australia or elsewhere, Williamson's had a policy at that time of preferring to cast lead players that they could bill as "direct from the West End", even if unknown; it was felt that an actor with that billing would always attract larger audiences than an Australian. Bailey also had a helpful resemblance to Rex Harrison who had created the Higgins part in London and New York, on record and in the eventual film of the work. Bailey, like Harrison, was not a singer; like Harrison, he handled the semi-spoken songs adeptly.