The Queen’s 83-year-old cousin was driving a Jaguar when he collided with another car in Sussex earlier this month.
Olivia Fellows, 21, claimed her Mini was written off in the incident on 2 June.
She said she was “in shock” and “completely stunned”.
The student told the Daily Mail: “I was driving north at 60mph and suddenly the Jaguar pulled out in front of me.
“He shot across the road, but it was so close that if I hadn’t emergency braked I would have gone straight into the middle of his car.”
Ms Fellows said her vehicle went into a spin before hitting the central reservation. “I clipped the curb and buckled the wheel arch of my car. The airbags went off and there was smoke coming from the engine,” she said.
Buckingham Palace said in a statement: “A royal household vehicle was involved indirectly in a collision. All the relevant details have been passed to the police.”