Born in Newcastle, Josephs had his first music studies with Arthur Milner in 1947, before qualifying as a dentist in 1951. After doing his military service as a dental officer in Germany, he studied music at the Guildhall School in London and in 1958, won a Leverhulme Scholarship, which enabled him to go to Paris and study with Max Deutsch. In 1962, he won first prize in the Jeunesses Musicales competition with the Comedy Overture: The Ants and a year later made a significant breakthrough when his Requiem in memory of the Jews who died during the Holocaust won the first International Composing Competition of the City of Milan and La Scala - then the biggest musical award in the world.

Although he kept a close eye on the work of his peers, Josephs never tried to follow mainstream trends. He learnt a great deal about serial techniques from Deutsch, himself a pupil of Schönberg, and used them repeatedly in his own music. However, he always retained a belief in expression through melody and this can be traced through his whole output. During the 1960s, Josephs became involved in writing music for film and achieved perhaps his most popular success with the score for the celebrated series, The Great War. He continued composing serious concert music - the Beethoven Variations for André Previn and the London Symphony, for example - and concentrated increasingly on writing music for ballets and opera: Rebecca, based on Daphne du Maurier's novel, was commissioned by Opera North and Cyrano by the Royal Ballet. His joie de vivre and raucous sense of humour are perhaps best reflected in the four works he wrote for the Hoffnung concerts, each based on a cartoon of the famous illustrator. He also composed Aelian Dances based on Newcastle tunes.
A concert of his music was played at Newcastle University in October 1996 in his presence.