Greenwood attended Liverpool Art School and the Central School of Arts and Crafts, and then started working at the Oxford Playhouse, in charge of the costume department.
Born in Liverpool, England, she works both in England and the United States. She started working in the Ray Diffen costume shop in New York City in 1962. Greenwood's work includes designing for over 100 productions, The Ballad of the Sad Cafe (1963), her first Broadway play, to Hamlet with Richard Burton (1964), 70, Girls, 70 (1971), Romantic Comedy (1979), The Sisters Rosensweig (1993), and Stephen Sondheim's Passion (1994).
In England, she was nominated for the Olivier Award for her costume designs for She Loves Me (1995). Her television work includes several Public Television plays for "The American Playhouse", made-for-television movies, such as In the Gloaming, HBO (1997) and the miniseries Kennedy (1983). For opera, she has designed for the Metropolitan Opera House, such as Ariadne in 1987.
For dance, she designed the original costumes for the Alvin Ailey dance Night Creature (1974). She teaches at the Yale Drama School. The executive directors of the Broadway League and American Theater Wing said, in part "She has made a significant imprint on the history of Broadway with her artistry.
Her work has not only elevated the craft of costume design but has inspired generations of designers to come.".
Married Ben Edwards, children: Sarah, Kate.