Arkwright was commissioned into the 12th Royal Lancers in 1924. He served in the Second World War as Deputy Assistant Adjutant General for 1st Armoured Division in France, as a General Staff Officer with 8th Armoured Division and as Brigadier responsible for Armoured Fighting Vehicles in the British Eighth Army before being made Commander of 23rd Armoured Brigade in 1943. As Commander of the 23rd Armoured Brigade he took part in the Sicily landings, the Italy landings, both part of the Italian Campaign, and later the Civil War in Greece. After the war he became General Officer Commanding 2nd Infantry Division in 1946, Director of the Royal Armoured Corps in 1947 and General Officer Commanding 56th (London) Armoured Division in 1948. His last appointment was as General Officer Commanding 7th Armoured Division from 1949 until his retirement in 1951.