Day was born in London, England but raised in Ilford. Day has stated that she came from “a very normal family with very normal jobs” as none of her family had ties to the entertainment industry. When five years old, she became passionate about acting after being taken to see ballets by her father, and joined Stagecoach Theatre Arts at the same age. During her formative years, Day and her family frequently shifted residences; in 12 years, Day studied in 11 different schools. Day attended the Sylvia Young Theatre School, and later the Arts Educational Schools, London (ArtsEd).
Day dislikes playing stock characters such as “the girlfriend of the main character or the girl next door”; she has cited her preference for roles requiring more action, including femme fatales. Day is trained in Mixed martial arts, boxing and Muay Thai. An aspiring writer, she wrote a psychological thriller novel and intends to convert it into a screenplay. Day is also a trained swimmer, and began preparing to participate in the 2012 Summer Olympics, but decided to leave swimming to focus on her acting career. She also knows surfing and jet skiing. Day has chosen not to comment on her sexuality; she stated in 2017, “Being straight, bisexual or gay is NOT a choice. All I’ll say on that matter.“