Gill told the Birmingham Mail that as she changed lanes she was hit by a 4x4 police car, but when she pulled over to swap insurance details the vehicle didn't stop.
Walker, 59, then alleges that a second 4x4 stopped alongside her Peugeot 107, and officers told her the other car couldn't stop because there was a "royal on board".
Ms Walker now claims that the police are denying liability and have left her with a large bill.
Ms Walker told the newspaper: "I asked 'Who's on board?' but he said 'I'm not obliged to say'. They took pictures of my car and said it was a police matter now.
"The police are saying it's all my fault. But the dashcam footage shows I've clearly got into the lane and they have hit me.
"It's like they've put all the blame on me and they've got solicitors involved. It looks like it's going to court because the police don't want to know."
Sophie, who is the wife of Prince Edward, was reportedly in Birmingham at the time of the alleged incident to open the first military and civilian wound research centre of its kind at the Queen Elizabeth Hospital.
A police statement said: "A West Midlands Police vehicle was involved in a collision with a third-party vehicle during a police operation, resulting in damage to both vehicles as they travelled along a three-lane carriageway.
"Due to operational reasons and force protocol, the vehicle did not stop, but another vehicle in convoy behind the damaged car did stop and details were exchanged with the third party for insurance purposes.
"A report has been completed by the driver, a police sergeant, and is with the force's insurers for assessment."
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