Thank you all for your interest. This is the final update, and I am sorry if anyone is disappointed by what I de decided.
The police station in my local town, four miles from where I live, is open only 9:30-12:30. But as I was in town this morning, I visited it. The constable on duty's first response was to say that I should report the incident to the police force of the area where it occurred, South Yorkshire. I explained that force had told me I must report it in person at a police station. She looked up things on computers, tried to get hold of the county control unit, went upstairs for advice. When she returned to the desk, she asked if I was prepared to make a formal statement and explained that could lead to my being summonsed as a witness. I decided the answer was no. We left it that she has logged my report of the incident.
Why am I not prepared to take it further? I cannot be precise about the time or place we saw this. I was the driver and so was concentrating on driving, not watching the daft lass. My passenger is not prepared to go to court. The car has no dash cam. We are not the kind of people who automatically pull out a mobile and record events. So it would be my word against hers, and any case would take place about a hundred miles from where I live. I was told that if the incident had occurred in my county, it would have been passed to the Traffic branch. Each police force does things differently. If the car remains untaxed, there will be automatic follow-up.
Many of you may think you would have continued to try to take this further.
I spent about an hour on Monday online trying to find out how to report it and checking I had the car details correct. Another hour or so yesterday checking online and working my way through phone menus to try to report it. A good half hour this morning at the police station. The time I spent cannot be recouped: tasks that should have been begun this morning (preliminary adjudication of competition entries from 200 or so local children, and admin to do with a volunteer group) have been pushed back.
However, if any of you have a young friend or relation in the Doncaster area who drives a blue Mazda, registration number EA60 UCL, do tell her to get her nearside rear brake light fixed, to tax the car and to keep her hands on the wheel and eyes on the road.