I wouldn't be surprised if neither child was in the car for the simple reason that there would have to have been at least one security officer in the car as well.
I believe that once George turns 12 he won't be able to fly with William but otherwise will be able to travel with him. Diana and Charles fought for that with William so that he could fly to Australia with Charles. After William turned 12, except for when they flew back from Switzerland after the Queen Mother's death, William flew on a separate plane to Charles while Harry would fly with Charles with no problems.
The window was down in the backseat of the car that followed them out so if the kids were in there they would have been visible. There was defintley a man in the back passenger side.
Knowing how weird the horse is there was likely a third car we didn't see.