The son of a local Surrey councilor and Justice of the Peace, Paul Mellows served as a flight-lieutenant with the RAF during World War II, and was awarded the Distinguished Flying Cross (DFC) in March 1945 for commendable skill and courage as a pilot while serving as a night fighter pilot with of the RAF Volunteer Reserve (RAFVR), 169 squadron. Mellows was educated at Monkton Combe School in Bath, and in 1939 was in the Monkton eight that had a surprise win over Shrewsbury School at Henley.
After the War, Mellows went to Clare College, Cambridge to study for a law degree. He became a double Blue in 1947 and 1948, when he was bow in the eight that won both races, the latter being in a record time and was the first winning boat to cover the course in under 18 minutes. Mellows was in the Leander crew trying to retain its Grand Challenge Trophy at Henley in 1947, but they were eliminated by the eventual winners, Jesus College, Cambridge. Mellows was one of three Old Monktonians in the Great Britain rowing squad at the 1948 London Olympics, Michael Lapage and Ran Laurie being the others. All three won medals, with Lapage and Mellows winning silver in the eights, and Laurie taking gold in the coxless pairs.