Miles William Arthur Peel Graham was born in Colchester, Essex, on 14 August 1895, the son of Major Henry Graham and his wife Ellen Peel, the great-niece of Sir Robert Peel. His mother later became Baroness Askwith through her second marriage to George Askwith, 1st Baron Askwith. He had a younger brother, Henry Archibald Roger Graham. He was educated at Eton and the University of Cambridge.
When the Second World War broke out in September 1939, Graham returned to his old regiment, which was now the Life Guards, with his old rank of captain. A wartime merger saw the Life Guards form part of the 1st Household Cavalry Regiment, one of the regiments of the 1st Cavalry Division, with Graham as its adjutant. Still a mounted formation, the 1st Cavalry Division moved to Palestine in 1940, and Graham became a staff officer at division headquarters. He then joined the headquarters of the newly-formed Eighth Army, where he served in the Q Branch under the Deputy Adjutant and Quartermaster General, BrigadierSir Brian Robertson. For his services during Operation Crusader, Graham, now a major and acting lieutenant colonel, was mentioned in despatches on 16 April 1942, and he was made an Officer of the Order of the British Empire on 9 September 1942. Robertson and Graham survived the purge of the Eighth Army staff that followed the arrival of Lieutenant General Bernard Montgomery as army commander in August 1942. Graham succeeded Robertson as DA&QMG when the latter became the Eighth Army's chief administrative officer. Robertson and Graham developed the concept of the Field Maintenance Centre for the support of fast-moving mobile operations. An FMC was like a military shopping centre, supplying all the needs of a corps. It dramatically reduced the response time to administrative needs of front line units and reduced the amount of paperwork that they needed to do to get their needs fulfilled. The FMC would later become a feature of British logistics in the Normandy Campaign. Graham succeeded Robertson as chief administrative officer of the Eighth Army, with the rank of brigadier in March 1943, and was mentioned in despatches on 24 June 1943. On 28 June 1943, he married Irene Lavender Francklin, the widow of an Army officer who had been killed in the fighting. Graham was made a Commander of the Order of the British Empire on 14 October 1943 for his role in the Allied invasion of Sicily. When Montgomery was appointed the commander of the 21st Army Group in December 1943, he took only seven officers from the Eighth Army with him, the most senior being his Chief of Staff, Major General Freddie de Guingand; Graham, his chief administrative officer; and Brigadier George Warren Richards, his Armour officer. Montgomery also requested, and eventually secured, the service of Brigadier R. W. Lymer. Graham was promoted to major general on 15 January 1944, and became the Major General Administration at 21st Army Group, with Brigadiers Randle Feilden, L. L. H. McKillop and Cyril Lloyd as his deputies. Graham was made a Companion of the Order of the Bath on 29 June 1944, and a Knight Commander of the Order of the British Empire on 5 July 1945. He was mentioned in despatches on 9 August 1945, and 4 April 1946. He also received some foreign awards, being made a commander of the United States Legion of Merit on 15 March 1945, and a Knight Grand Officer of the Netherlands Order of Orange Nassau with Swords on 20 January 1947. He relinquished his commission on 28 March 1946, and was granted the honorary rank of major general. He retired on 6 February 1947.