Heaton was born in Leeds, West Yorkshire, and raised on a council estate in Bridlington by his mother. He moved to London when he was 16 and joined London noise-rock band Comanechi as the drummer. He toured with them for more than a year. Heaton then joined London-based psychedelic band Half Loon.
In 2014, Heaton played a role in Life Needs Courage, a short film. In 2015, he made his acting debut with the ITV crime drama series DCI Banks, playing the role of Gary McCready. He appeared as Riley in ITV's detective series Vera. He guest-starred in BBC One's medical drama series Casualty as Jason Waycott. In 2016, he appeared in the thriller film Shut In, co-starring Naomi Watts and Oliver Platt, and directed by Farren Blackburn. Heaton also plays Jonathan Byers in the Netflix supernatural drama series Stranger Things. In 2016, he starred in the drama film As You Are.
In May 2017, Heaton was cast to star as Samuel "Sam" Guthrie / Cannonball in The New Mutants, a 2020 film based on the Marvel Comics comic book of the same name. In August 2018, the BBC announced that Heaton would be playing Joseph Merrick, commonly known as the "Elephant Man", in a new two-part drama. The casting drew criticism from a disability charity group, the group's head saying that not using a disabled actor was a missed opportunity.
