Glenister was born in Harrow, Middlesex, and grew up in Hatch End. He is the son of director John Glenister and Joan Glenister, and the younger brother of fellow actor Robert Glenister. He is of Welsh ancestry from his maternal side. He attended Hatch End High School, and with the encouragement of his then-sister-in-law Amanda Redman, he pursued acting and attended drama school at the Central School of Speech and Drama.