Morgan McElligott studied at University College Dublin, earning a medical degree. While in school he was a member of the boat club that won the 1947 All-Ireland Senior Rowing Championship. He began his full-time medical career in Dublin, before moving to London in 1950. After six years working in England, McElligott moved back to Ireland and took a position in 1956 as a consultant at Portiuncula Hospital, and would then work at the facility for the next 33 years. He specialized in cardiology and helped establish the first cardiac care unit Ireland at Portiuncula.
In 1989 McElligott retired from Portiuncula and he and his wife moved to Baghdad, where he helped develop the Ibn Al-Bitar Hospital. After 18 months they left, with the first Gulf War imminent, moving first to Cyprus and then to a hospital in Tabuk, Saudi Arabia where he served as chief of internal medicine for the Royal College of Surgeons. McElligott finally fully retired after 18 months in Saudi Arabia. He moved back to Ballinasloe, Ireland with his wife, although they later settled in Coosan West. In retirement he was involved with the East Galway Hunt Club, serving as their President, and with the County Roscommon Hunt. McElligott was also involved in charity initiatives, such as generating funds that allowed the purchase of a CT (computed tomography) scanner for the department of radiology at Portiuncula.