Beazley was born in Kensington, London. He began acting professionally as a teenager in the early 1930s, including appearances with John Gielgud in productions of Hamlet (1934) and Romeo and Juliet (1935). After serving in World War II, he owned an antique shop for several decades. At the age of 75, Beazley returned to acting, and subsequently appeared in a variety of stage and film productions, as well he became a painter in his nineties. A 2010 article by Nicholas de Jongh in The Independent described "the amazing" Beazley as "the last theatre survivor of his generation". Beazley died on 12 June 2017 at the age of 101.