As a boy, he learned to play the violin and piano and sang in a choir. He studied at the University of Birmington (1941 - 1944) and the Royal College of Music (1944-1946). His instrument was the horn, but he also studied theory and composition. He took a position as a lecturer at Aberdeen University. In 1948 he won the Carnegie Scholarship, which enabled him to study with conductor Herman Scherchen. His first conducting job was with the Carl Rosa Opera Company, beginning in 1950. In 1952 he moved over to Covent Garden. He first attracted wide attention with a performance of Der Freischütz in April 1954. Later that year he had an even more dramatic performance when he took over at the last minute (and without an orchestral rehearsal) a new production of Les contes d'Hoffmann.
In 1963 he was selected to lead the first Western performances in nearly 30 years of Shostakovich's once-banned opera Lady Macbeth of Mtsensk, in its new edition renamed Katerina Ismaîlova. Moreover, he prepared the English translation used in that production. He went on to make performing translations of other Russian operas such as Mussorgsky's Khovanshchina and Prokofiev's War and Peace. In 1966 he was appointed assistant to the music director, Georg Solti, at Covent Garden. In 1967 he conducted a full cycle of Wagner's Ring, the first British conductor to do so since Beecham. He officially resigned his staff position at Covent Garden at the end of the 1968-1969 season, but frequently conducted there, including premieres of Richard Rodney Bennett's Victory in 1970 and Peter Maxwell Davies' Taverner in 1972.
In 1969 he was appointed musical director of the Australian Opera. There he conducted the first opera performed in the new Sydney Opera House on September 28, 1973, War and Peace. His first appearance with the Welsh National Opera was in 1975. He gave up his Sydney position in 1976. He also conducted the Netherlands Radio Orchestra until 1983. In 1980 he was appointed principal conductor of the BBC Northern Symphony Orchestra in Manchester, which later the same year was renamed the BBC Philharmonic. When he left the orchestra in 1991, he was given the title Conductor Emeritus. He was knighted in the same year.
His scholarship in the field of Russian music has resulted in a completion of Prokofiev's opera Maddalena from the composer's manuscript sketches, and the preparation and premiere recording of incidental music by Prokofiev to Pushkin's Eugene Onegin. He regularly recorded for the Chandos.
Downes and his terminally ill wife, a former ballet dancer whom he had married in 1955, died in Zurich, Switzerland, in July 2009.