Gillan was born and grew up in Inverness, Scotland. She learned to play the piano when she was seven. She started acting when she was very young. She joined several local youth theatre groups. She also acted at her school, Charleston Academy. When she was sixteen, Gillan went to Edinburgh's Telford College to study acting. Later, she studied at the Italia Conti Academy of Theatre Arts drama school in London. Gillan moved to London when she was 18 to study at drama school.