Born in Harrison Hot Springs, British Columbia, Elliot was educated at Queen's University where he graduated with a medical degree in 1937.

[1] He completed post-graduate training at The Hospital for Sick Children in Toronto, from 1943 to 1944.

[1] Elliot joined the Royal Navy Volunteer Reserve in 1940 as a medical officer, where he served on escort ships in the North Atlantic.

In June 1942 he transferred to the Royal Canadian Navy Volunteer Reserve and served ashore at the hospital at HMCS Naden in Esquimalt, British Columbia.

Allan was given the title of Queen's Honorary Surgeon by the Medical Household.