Although her father was a professional cellist, Florence Hooton did not receive lessons on the instrument until she was nine years old. At that point, her father enrolled her with his own teacher, Warren Evans, who was well-known as a chamber player. She stayed with him until she was 14, when she began to attend the London Cello School. Her teacher there was one of the leading cellists in the British Isles, Douglas "Duggie" Cameron. After she won a scholarship at the Royal Academy of Music she continued studies with Cameron.
She made a notable debut recital at London's Wigmore Hall in 1934. She had founded a trio with Frederick Grinke and Dorothy Manley, and with them played her first orchestral appearance, at the Proms with the BBC Symphony Orchestra under Sir Henry Wood, in the Beethoven Triple Concerto.
She took an interest in modern British music, and soon gave the world premieres of Gordon Jacob's Divertimento for Unaccompanied Cello and the Legend-Sonata by Arnold Bax, with Bax's favored pianist, Harriet Cohen.
Even so, she continued her studies, going to Zurich, where Emmanuel Feuermann had settled to teach. She recalled Feuermann privately accomplishing the remarkable feat of playing Mendelssohn's violin concerto on cello, not transposed or lowered an octave, but at the original violin pitch. Like most cellists who studied with Feuermann, she selected one of many particular things about his technique and adopted it into her own. In her case it was a knack of putting the bow on the string at exactly the right angle so that it started to vibrate immediately." She has tried to pass that on to her students.
During one of her performances in 1934 present in the audiences were the composers Frank Bridge and his pupil Benjamin Britten. Bridge noticed that the piece she was playing had a large proportion of very high notes and whispered to Britten, "If that girl gets one more of those top notes spot-on I'm going round to ask if she'd like to give the first performance of my cello sonata." The work in question had been turned down by two leading British cellists without changes to its high notes. Hooton worked on it for two years, working out practical fingerings for this work once called unplayable, and in 1936 did successfully premiere the concerto, which is called Oration.
She had a successful playing career in Britain. She became the first cellist to play on television. This was a program in which she played the Haydn D Major Concerto while two unknown ballet dancers, Margot Fonteyn and Robert Helpmann, interpreted it. She played the premieres of several important British works, including concertos by Alan Bush, Jacob, and Leighton.
In 1950 she formed a trio with violinist David Martin (her husband) and pianist Iris Loveridge. The ensemble stayed in business for over a quarter of a century, finally disbanding in 1976. During all this time she also pursued a teaching career, and in 1964 became professor of cello at the Royal Academy of Music. This was a result of an observation by her and her husband that while both of their instruments had many excellent players, there were not that many really good teachers. Both agree that investing their energy into pupils and then seeing results gives "all the compensation you need."