Hambro was born on 24 July 1930.
He was an heir to the Hambros Bank. His great-great-grandfather, Carl Joachim Hambro, was an immigrant to England from Denmark who founded the Hambros Bank in 1839. His father, Charles Jocelyn Hambro, was a merchant banker. His mother, Pamela Hambro, died when he was 21 months old. He grew up at Delcombe Manor in Dorset.
After his mother's death, his stepmother became Dorothy Mackay, who was banker Marcus Wallenberg's ex-wife. During World War II, he was sent to live first with the Wallenbergs in Stockholm and later with the Morgans, another banking dynasty, in New York City. He returned to England in 1943.
He was educated at Eton College, where he played on the cricket team. He then served in the Coldstream Guards for two years.
Hambro started his career at the family business, Hambros Bank, in 1952. He was appointed managing director in 1957, Deputy chairman in 1965, and chairman in 1972. He was in charge through interesting but turbulent times, beginning with the stock market and property crash of 1973–74. Hambros was one of the leading banks called in by the Bank of England to launch the financial lifeboat which dealt with the collapse of the Slater Walker empire and saved the financial system from collapse. In 1998, Hambros Bank was acquired by Société Générale. It represents its private wealth management subsidiary, SG Private Banking.
From 1987 to 1999, he served on the board of directors of the shipping and distribution group P&O. He also served on the Boards of the Guardian Royal Exchange Assurance and Taylor Woodrow.
Hambro served as the senior honorary treasurer of the Conservative Party from 1993 to 1997. He was in charge of replenishing the £19 million overdraft. His efforts were rewarded with a life peerage, created 26 September 1994; he took the title Baron Hambro, of Dixton and Dumbleton in the County of Gloucestershire.
He served on the Board of Trustees of the British Museum from 1984 to 1994.
Hambro married his first wife, Rose Evelyn, the daughter of Sir Richard Cotterell of the Cotterell baronets, in 1954. They had a daughter Clare Evelyn (married to Eivind Rabben) and two sons, Charles Edward (divorced from Nicole Nicholas) and Alexander Robert (married to Hattie Ward Jones). They were divorced in 1976. In the same year he married his second wife, Cherry Huggins, daughter of Sir John Huggins, a former Governor of Jamaica. She was a divorcee with one daughter, Miranda. He had nine grandchildren: Christiana, Tatiana, Charles, Edward, Alexander, Ben, Marina, Jemima and Sam.
He was the owner of two manors in Gloucestershire: Dixton Manor in Alderton, and Dumbleton Hall in Dumbleton. He organised pheasant shoots on the latter estate. He was a member of the Marylebone Cricket Club. He also gambled at the Bahamian Club in Nassau, Bahamas.
He died on 7 November 2002 in London.