Julie Andrews received no criticism for her cockney Eliza. Audrey Hepburn did. Julie received universal raves for her Eliza.
Julie Andrews was offered director and cast approval from Jack Warner to appear in CAMELOT, against Logan's wishes. Logan did not want her, but she could have had him replaced. When Burton declined, she had no interest in doing the role.
Finally, the shooting schedule for POPPINS was such that all of the principal photography was completed, prior to MY FAIR LADY's principal photography (due to all the effects and animation that had to be added). She could have done both. MFL's schedule conflicted with THE AMERICANIZATION OF EMILY and THE SOUND OF MUSIC; and yes, Disney waited for her because she was pregnant.
Oh, and to reply to the OP's question, God no!!!
Updated On: 12/16/13 at 11:00 PM