Born in Hausham, Wurmheller was a glider pilot in his youth, and volunteered for military service in the Luftwaffe of Nazi Germany in 1937. Following flight training he was posted to Jagdgeschwader 53 (JG 53—53rd Fighter Wing) in 1939. He claimed his first aerial victory on the Western Front on 30 September 1939, a phase of World War II dubbed the Phoney War. He then served as a flight instructor before returning to JG 53 for the Battle of Britain. In June 1941 he participated in Operation Barbarossa, the German invasion of the Soviet Union. He was awarded the Knight's Cross of the Iron Cross on 4 September 1941 after 32 victories. Following another tour as fighter pilot instructor, he was posted to Jagdgeschwader 2 "Richthofen" (JG 2—2nd Fighter Wing) operating on the Channel Front. Wurmheller claimed seven aerial victories during the Dieppe Raid on 19 August 1942 and after his 67th victory was awarded the Knight's Cross of the Iron Cross with Oak Leaves on 1 October 1942.

In April 1943, Wurmheller was appointed Staffelkapitän (squadron leader) of 9. Staffel (9th squadron) of JG 2 "Richthofen". On 8 June 1944, two days after the Allied Invasion of Normandy, he was given command of III. Gruppe (3rd group) of JG 2 "Richthofen". He and his wingman were killed in a mid-air collision during combat near Alençon, France, on 22 June 1944. Wurmheller was posthumously promoted to Major (major) and was awarded the Knight's Cross of the Iron Cross with Oak Leaves and Swords on 24 October 1944.