Coates was born in Reigate, Surrey, England, the daughter of Kathleen Voase (née Rank) and Major Laurence Calvert Coates. Her first passion was horses. As a girl, she thought she might become a race-horse trainer.
Coates attended the Reigate village school called the Micklefield School. She then attended High Trees School in Horley (Surrey). She graduated from Bartrum Gables in Broadstairs (Kent).
Coates was at the centre of a film industry family. Besides being the niece of J. Arthur Rank, she was married to the director Douglas Hickox for many years. Her brother, John Coates, was a producer (The Snowman and Yellow Submarine), and her two sons, oldest Anthony Hickox (b. 1959) and youngest James D. R. Hickox (b. 1965) used to be directors, and her daughter Emma E. Hickox (b. 1964) is also a film editor.
"In a way, I've never looked at myself as a woman in the business. I've just looked at myself as an editor. I mean, I'm sure I've been turned down because I'm a woman, but then other times I've been used because they wanted a woman editor." 
"...I guess I've been lucky that most of the time I've been in the same direction as the director. I try to work with directors whose work I like and find interesting. When I was younger, I had to find work where I could, and I had some not great experiences with directors."
"You have the courage of your convictions. When you're editing you have to make thousands of decisions every day and if you dither over them all the time, you'll never get anything done."
"I seem to get the rhythm from the performances. I like to feel I'm very much an actor's editor. I look very much to the performances and cut very much for performances rather than the action. I think that's important, what's in the eyes of the actor."
1947: The End of the River – second editor (uncredited)
1949: The History of Mr. Polly – assistant editor (uncredited)