He was part of a well-known English family of musicians that included several professional singers and organists during the 17th and 18th centuries.
Born into a musical family, his initial studies were with his father alongside his younger brother Charles Young, who would also become a successful organist and minor composer.
Music historian Charles Burney wrote that Anthony was organist at St Katherine Cree from 1702 to 1706, but modern scholarship makes this seem unlikely.
His niece Cecilia Young (1712–1789) was one of the greatest English sopranos of the 18th century and the wife of composer Thomas Arne.
Young's nephew, Charles, was a clerk at the Treasury, whose daughters, Isabella, Elizabeth, and Polly followed in the foot steps of their aunts to become successful singers.