Jackson-Cohen was born in Westminster, London. His mother, Betty Jackson, is an English fashion designer, and his father, David Cohen, who is French-Israeli, is his mother's business partner. Because of his French heritage he attended French school at the Lycée Français Charles de Gaulle in London and claims to have a slight French accent.
On leaving school, Oliver was advised to treat acting as a hobby rather than a career. After a role in a play at the Lyric Theatre got him an agent, he began attending auditions and put university on hold to apply for a place at drama school in New York. Shortly afterwards he received his first role.
He also studied with Michael Luggio at the Malisa Theatre Academy. "I can safely say that I have never met anyone quite like Michael Luggio. His approach to acting and teaching is incredibly invigorating. He breaks all the barriers down, so you are left with the simplest and most honest performance possible. His classes are built around making the safest environment for his actors, in order for them to discover, learn and rid themselves of fear. It becomes a platform where anything is possible. Michael has helped me more than anyone I have ever met. He continues to bring it back to the craft and makes you feel safe and confident as you make your way out into the industry." He attended the Youngblood Theatre Company at weekends, and at 15 landed a small part in Hollyoaks. After school, he began a degree in French literature lasting "for about two weeks. I hated it." He soon got a job washing vases for a florist to keep him going between auditions. "It wasn't great but when you're that broke you'll do anything.