His mother, Mary Turner, was a keen musician who played piano and violin and sang in Charles Sanford Terry's Bach Choir in Aberdeen.
Bate had intended to continue his studies as a postgraduate in geology but having been a member of the university's dramatic societies and with the new drama department of the Aberdeen station of the BBC frequently choosing him for amateur cast broadcasts, not least for his English accent, he applied for and was appointed to a post with the corporation in London.
Bate spent the majority of his career working for the BBC's music department — starting as a balance control assistant between 1934 and 1937 and then as a studio manager from 1937 to 1939.
Whilst in London he would frequent the Caledonian Road, Portobello, and Bermondsey markets, making friendships with those who shared his interests, such as Canon Francis Galpin, who encouraged Bate to turn his scientific education to the study of musical instruments.
Bate used his carpentry skills to make and restore instruments in his collection and after learning metalworking techniques, made reproductions of draw-trumpets used by David Munrow's Early Music Consort of London.
As well as writing articles for the Grove Dictionary of Music and Musicians, Bate wrote the books The Oboe (1956), The Trumpet and Trombone (1966), and The Flute: a Study of its History, Development and Construction (1969).