Weedon was born in Burges Road, East Ham, London. He began learning classical guitar at the age of 12, and decided to become a professional musician. In his teens during the 1930s, he led groups such as the Blue Cumberland Rhythm Boys, and Bert Weedon and His Harlem Hotshots, before making his first solo appearance at East Ham Town Hall in 1939. He worked with leading performers including Stephane Grappelli and George Shearing, and performed with various big bands and orchestras, including those of Ted Heath and Mantovani.