From the age of 7 he was taught by his father and he won an open scholarship to the Royal College of Music, becoming a pupil of Charles Draper. During the 1920s he played with the orchestra of the Royal Opera House and the BBC Wireless Orchestra before becoming principal clarinettist of the newly formed BBC Symphony Orchestra. He left the BBC Symphony Orchestra in 1946 to concentrate on chamber music.
He was also principal clarinet of The Philharmonia Orchestra and can be heard on the Toscanini recording of the Brahms Symphonies. Thurston can also be heard on Volume 1 of Historical Clarinet Recordings on the Victoria Soames Samek's Clarinet Classics CD Label.