The company said Haig was stepping down by mutual consent and with immediate effect.
The designer is planning to stage “happenings” four times a year instead of shows in February and September.
Theresa May was photographed for the magazine by Annie Leibovitz.
Madison Avenue windows will be dedicated to British brands for two weeks.
Wintour will be given the title of Dame, her second honor.
Fellow Conservative Nicky Morgan has accused May of profligacy in her Amanda Wakeley purchase.
Annual ceremony also saw Charlotte Tilbury, Gary Card scoop prizes.
The British accessories and ready-to-wear designer returns to the city with a new store on the buzzy corridor.