Brady, 48, who has been a non-executive director of Taveta since 2010, has assumed the role of non-executive chairman with immediate effect, succeeding Anthony Grabiner, who has been with the company for 15 years.
Taveta owns the Arcadia business, which runs fashion retailers Topshop, Topman, Wallis, Miss Selfridge, Dorothy Perkins, Evans, Burton and Outfit.
Arcadia employs over 24,000 people and has an annual turnover in excess of 2 billion pounds.
Green's reputation was damaged after he was blamed by British lawmakers last year for the demise of the BHS department store chain.
He owned BHS for 15 years before he sold the loss-making retailer to Dominic Chappell, a serial bankrupt with no retail experience, for one pound in 2015.
In February, Green paid 363 million pounds to plug a hole in BHS's pension schemes.
However, some lawmakers still want Green to be stripped of his knighthood, awarded in 2006 for services to retail.
“It is a privilege to have been invited to chair the board and I look forward to working with my colleagues as we concentrate on driving the Arcadia brands forward on their global expansion," said Brady.
She was ennobled in 2014 by the then Prime Minister David Cameron.
(Reporting by James Davey; Editing by Mark Potter)