Donald Core was the son of T.H. Core, professor of physics at Owens College Manchester. He was educated at the University of Manchester, and qualified MB ChB in 1906, followed by a MD in 1910. He was a house physician at MRI and did postgraduate work in Paris, working at the Pasteur Institute in 1912. From 1912-18 he was honorary physician at Ancoats Hospital Manchester, and was elected honorary assistant physician at MRI in 1917. Core was a lecturer in clinical medicine and neurology at the University of Manchester. Core served with the RAMC during the 1914-18 war in Mesopotamia, where he developed a debilitating illness.
Core was a neurologist, and wrote two books: Functional nerve disorders (1922) and Examination of the central nervous system (1928). He was noted for the degree of attention he gave his patients, and was considered an effective teacher.