Her father Bjorn was a businessman from Norway; her mother, Mia Prodan, was from Romania. Both were from theatrical families. Björnson grew up in London; she studied at the Lycée Français, and then at the Byam Shaw School of Art and at the Central School of Art and Design.
She designed sets and costumes for theatre, ballet and opera. She worked for the Royal Shakespeare Company, and designed Andrew Lloyd Webber's The Phantom of the Opera, which won her the Tony Award for Best Scenic Design, the Drama Desk Award for Outstanding Set Design, and the Drama Desk Award for Outstanding Costume Design, and the Trevor Nunn production of Aspects of Love.
Björnson was course director for Theatre Design at the Central School of Art and Design.