The oldest child of Christopher Drew, a cabinet-maker, and Jane (née Church), a pediatric oncology nurse, Hannah was born in the heart of downtown Toronto. Seeking a quieter place to raise a family, the three moved to London, where they were later joined by Hannah's younger brother Harrison (Harry) Drew.

Hannah attended Wilfrid Jury Public School, and would often accompany her mother to rehearsals for community theatre productions where Jane volunteered as a costume designer. It was at the age of five, during a rehearsal for The Cemetery Club, that Hannah fell in love with acting. While taking a break from the long hours of cue-to-cue, the experienced ensemble of actors invited Hannah on stage to recite a poem she had learned in class. Happily, and with no hesitation, Hannah clambered up onto the stage, stood center, and sang out her piece. After that, she was hooked.

She went on to study at Central Secondary School, winning lead roles in The Grand Theatre's High School Production several years in a row, first as Juliet (Romeo & Juliet), and then as Geraldine, the villainess in James Reaney's Listen to the Wind. She also participated in local Fringe Festivals, winning praise for her turns as Jo March, and Tess Durbeyfield. After high school, she pursued her passion at The University of Toronto, proving her strengths in both classical and contemporary theatre, and graduating at the top of her class, and landing her first agent.

She has since appeared in several TV shows, including Zoo, The Flash, and Supernatural. In 2015, she collaborated with her long time friend, Karen Slater, to write, produce, and star in the first series to portray Sherlock Holmes, and his faithful Watson, as women, releasing the project under the title of Baker Street.