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The Duke and Duchess of Cambridge's convoy was travelling from London to Windsor on Monday afternoon when the accident happened.
The woman, Irene Mayor, who is in her 80s, is in a serious but stable condition, police said.
The Sun newspaper, citing witnesses, said the escort was on the wrong side of the road clearing traffic for the couple's convoy when the incident happened.
Britain's police watchdog, the Independent Office for Police Conduct, said it was investigating.
The royal couple were "deeply concerned and saddened" to hear of the accident, according to a spokeswoman, and a member of their staff visited the woman and delivered flowers.
"Their Royal Highnesses have sent their very best regards to Irene and her family and will stay in touch throughout every stage of her recovery," the spokeswoman said.
The accident took place when the couple were on their way to Windsor for the St George's Chapel service commemorating the Order of the Garter.
In January this year, a woman was injured in a car crash involving Prince Philip near the royal family's Sandringham Estate.
The Land Rover that Philip was driving flipped over. The 98-year-old was uninjured and voluntarily surrendered his driver's licence soon after.