Greenwood was born in London. He learned piano and viola from his parents and at 18 entered the to study viola and horn.
He was a classical composer who also wrote music scores for nearly 50 films from the 1930s to 1950s. He will be found on the credits of films from to Grand National Night (1954). While he no doubt gained considerable satisfaction from these compositions - and access to a large audience - there were also frustrations as the film editing process frequently required the removal or addition of a bar quite regardless of the overall theme of the piece.
Whether his compositions of incidental music for Shakespeare"s Merchant of Venice and A Midsummer Night"s Dream were intrinsically more satisfying is not known.
Serious works include Louisiana Belle Dame Sance Merci, Pippa Passes, Puncinello, and Salute to Gustav Holst which was premiered at the Sir Henry Wood Promenade Concerts in 1936. During the War he worked on the staff of the British Broadcasting Corporation European Service as Assistant Music Supervisor.
He died at Ditchling, aged 75. Many of his manuscripts are held in the archives of McMaster University.