Goodwin was born in Plymouth in Devon. He was taught the piano at an early age, and studied the trumpet in London at the Guildhall School of Music. His first job in music was as a copier and arranger working for a variety of publishing companies and bands, including some attached to the British Broadcasting Corporation. He later worked as a conductor in recording sessions for a number of popular music artists, Petula Clark among them.
In the 1950s he joined Parlophone, and worked alongside George Martin there. He continued to accompany artists such as Peter Sellers as well as beginning to broadcast and release records with his own orchestra.
Goodwin's first film work was for documentary films, and his first feature was Whirlpool in 1958. He worked on a number of war films in the 1960s, including 633 Squadron and Operation Crossbow as well as the above mentioned. He also wrote the scores for four Miss Marple films, and two movies featuring Morecambe and Wise.
Goodwin was also a guest conductor with a number of symphony orchestras, including the Royal Philharmonic Orchestra, the City of Birmingham Symphony Orchestra and the Detroit Symphony Orchestra. He usually conducted film music by himself and others, light music, and arrangements of popular music on these occasions. He also made a number of records of orchestral versions of pop tunes.
Goodwin won three Ivor Novello Awards, including one for lifetime achievement in 1994. He was also given freedom of the City of London.
Goodwin was a sufferer of asthma. He died suddenly in 2003 at his home in Reading having recently completed conducting his last series of Christmas concerts with the Bournemouth Symphony Orchestra.