Henry was born in London, England, to Protestant nonconformist parents William Beddome and Eleanor Smith.
He migrated to Canada and became "Surgeon and Clerk" at York Factory in Manitoba for five years starting in 1852.
After this term, he traveled to the Red River Colony where he met and married Frances Omand.
Beddome and his family returned to Red River in 1864 where he established a practice and a permanent residence near St Andrew's Church.
Henry Beddome and fellow doctors, including Curtis James Bird and John Christian Schultz were the founders of the Medical Health Board of Manitoba which was incorporated in 1871 and became the College of Physicians and Surgeons of Manitoba in 1877.