Bellingham was born on 29 March 1955 in Cheltenham, Gloucestershire. He is the son of Arthur Henry Bellingham and his wife June Marion Cloudesley Smith. He was privately educated at Wellesley House School in the town of Broadstairs in Kent, followed by Eton College in Berkshire. He went on to study at Magdalene College, Cambridge, where he received a law degree in 1977. During his time at Cambridge, he was a member of Cambridge University Liberal Club and served as Joint Master of the Cambridge University Draghounds.