She attended the Royal College of Music in 1943. She established herself throughout the 1950s, winning the prestigious Queen's Prize in 1952, making her debut the following year at the Proms, the annual classical music series at London's Royal Albert Hall, and playing with notable musicians throughout Europe.
Ms. Fleming became a professor at the Royal College of Music.
In 1970, she stood in for Bette Davis in a film called ''Connecting Rooms,'' in which the Hollywood star played a cellist.
Her playing career ended in 1993 following a stroke, but she continued to teach.