"A lady was on the ground being attended to by the paramedics and a police motorcycle was lying on its side on the road nearby."
A spokesman for the royal couple told the Mail they were "deeply concerned and saddened" and had been in touch with the victim's family. Flowers have been sent.
The Independent Office for Police Conduct said the 83-year-old victim was taken to a London hospital where she is in a serious but stable condition.
"In line with procedure, the Metropolitan Police Service referred the collision to the IOPC," it said in a statement.
"Our staff attended the scene of the incident and after careful consideration, we have launched an independent investigation. The investigation is in its very early stages and the officer involved is assisting our enquiries as a witness."
The incident occurred on Upper Richmond Road, south-west London, at 12.50pm on Monday local time.
William and Kate were travelling to join the Queen for an Order of the Garter ceremony. When members of the Royal Family use public roads they normally use an official car – in this case most probably a Land Rover Discovery – preceded by another Land Rover with armed police.
The two cars have motorbike outriders to the front and the back, for security reasons and as a means of clearing a path.
It is believed that the incident involved one of the advance party of riders who were designated to hold back the traffic.
It is not yet clear whether Mayor was actually struck, or whether she fell. This is something the watchdog will investigate. The duke and duchess did not see the accident and, in line with security procedures, the convoy kept driving.
The couple continued with the engagement but were informed about what had happened as soon as possible and immediately set about contacting Mrs Mayor's family.
The fact that the IOPC has started an investigation does not mean the police officer involved has done anything wrong.
The body is responsible for investigating incidents involving officers and staff which result in a death or serious injury.
Its work is done independently of the police, government and any interest groups. The Mail understands William had been keen to visit the victim yesterday but felt that it could be too intrusive, and instead both he and Kate sent flowers, after initially making contact via a family liaison officer. A member of their team from Kensington Palace has also been to see family members to express their concern.
The couple have asked to be kept informed of Mayor's condition and the family have said they will stay in touch. Police sources said that any officers involved in an IOPC investigation would be prevented from getting in touch with anyone who had come to harm.
A Kensington palace spokesman said: "The Duke and Duchess of Cambridge were deeply concerned and saddened to hear about the accident on Monday afternoon.
"Their Royal Highnesses have sent their very best wishes to Irene and her family and will stay in touch throughout every stage of her recovery."
It is the fourth road accident involving a member of the Royal Family in seven months. The Duke of Edinburgh's smash at Sandringham earlier this year resulted him giving up his driving licence at the age of 97.
In March it emerged that a receptionist from Birmingham was locked in a legal fight over a repair bill for her Peugeot 107 after her car was hit by a police driver chauffeuring a royal, believed to be the Countess of Wessex, last November.
Last week it emerged that the Queen's cousin, the Duke of Kent, could face a police investigation following claims that the 83-year-old allegedly pulled out in front of another driver, causing her to crash her car.
