Algernon de Blois Hayes-Hill was born in Bulawayo, an industrial center in Matabeleland, a district of the self-governing British Crown colony of Southern Rhodesia (now Zimbabwe).
He began his dance training as a seven year old youth with Elaine Archibald, where his grandmother, tasked with his care during WWII, was the studio pianist.
Algernon's parents, Harry M Hayes-Hill and Una May (née Spurr) were of British and Canadian stock, supported him in his ambition to pursue a career as a professional dancer, and made it possible for him to travel to London in 1951.
Once in London, he enrolled in classes with Ruth French and, after three months' study with her, auditioned, successfully, for admission to the Sadler's Wells Ballet School.
In 1957, he created the role of King of the South in John Cranko's The Prince of the Pagodas, set to the music of Benjamin Britten, and danced a difficult variation in the act 3 pas de six.
There, appearing alongside leading dancers Marcia Haydée, Richard Cragun, and Ray Barra, Burne enjoyed considerable success.
For eighteen months, he danced in new productions created by Cranko for the company, including Daphnis and Chloē and Romeo and Juliet (both 1962), as well as in ballets in the existing repertory.
When renowned French ballerina Yvette Chauviré came to Johannesburg as guest artist, Burne danced as Albrecht to her famous interpretation of Giselle in April 1963.
Engaged to produce Jesus Christ Superstar, the rock opera by Andrew Lloyd Webber and Tim Rice, he returned to Salisbury, Rhodesia, in late 1971.