Baker was a jazz trumpeter, equally adept at the cornet and the bugle, who was lead trumpet for Ted Heath's big band in the 1940s. He went on to form his own group, the Baker's Dozen, and soon landed a BBC radio series. In 1954, Baker dubbed the trumpet playing for actress Kay Kendall in the film Genevieve, and performed in soundtracks for James Bond films and The Young Indiana Jones Chronicles. A colleague of Jack Parnell from the Ted Heath days, he became a permanent member of ATV's band beginning in the 1960s, and worked The Muppet Show for all five seasons.