In 1939, Bridgeman was petroleum advisor to the Ministry of Economic Warfare, and from 1944 to 1946 Principal Assistant Secretary to the Ministry of Fuel and Power. From 1960 to 1969, he was chairman of British Petroleum.
Bridgeman was honored as a Commander of the Order of the British Empire in 1946 and as a Knight Commander in 1964. He was also a Knight of the Most Venerable Order of the Hospital of St John of Jerusalem.
Bridgeman married Diana Mary Erica Wilson, daughter of Humphrey Minto Wilson, on 23 February 1933. They had four daughters.