The bystander said he saw Harry, 29, in the back of the car with his head in his hands, constantly looking backwards at the injured men with "concern etched on his face".
Last night the rider from the Special Escort Group was discharged from hospital, while the driver was said to be in a "stable" condition.
Prince Harry visited New Zealand House in London on September 8, 2014 to meet New Zealand athletes competing in the Invictus Games. Photo / Getty Images
The Metropolitan Police also began an investigation into the accident, which occurred at around 9.40am on Prince Albert Road, adjacent to Regent's Park and London Zoo.
A cyclist, who wishes to remain anonymous, saw the convoy of two blacked-out Range Rovers and four motorbike outriders pass him at "incredibly high speed".
He said: "I decided to speed up to see if I could catch up with them. I really wanted to know who it was.
Prince Harry at the opening ceremony of the Invictus Games in London on September 10, 2014. Photo / Getty Images
"I didn't see the actual crash but came across it just seconds after it had happened. There was debris everywhere, steam coming out of the car bonnet. It was carnage."
CCTV footage obtained exclusively by the Mail appears to show the minicab driving down a side street, stopping at the junction and then pulling right across the main road - straight into the path of the police motorbike, which was on the wrong side of the road.
The bike had its police lights on but there were no sirens.
The witness said: 'The first thing that struck me was the motorbike rider in his high visibility jacket lying prone on the ground. He had been put in the recovery position and had his helmet off.
"He was definitely the focus of everyone's attention. The motorbike was well and truly wedged in the bonnet of the Prius."
Not wanting to get in the way of the emergency effort, the cyclist began to ride off when he noticed the second black Range Rover parked down the road.
"I could see Prince Harry clearly in the back of the car with his head in his hands,' he said. 'He looked really distressed and was constantly looking back over his shoulder to see what was happening. It was clear that for security reasons he had to stay in the car."
Royal aides confirmed last night that Prince Harry's convoy had been involved in the crash but that he hadn't been injured.
A source close to Harry said: "It was clearly a shocking incident but he was pleased the officers were able to stop and administer important immediate first aid.' A source on the scene said: 'The police were fantastic and sped into action, assessing those who were injured and treating them until the ambulances arrived."