Whose insurance to use?

I was driving my wife's car and after reversing out of a very tight spot and whilst waiting for my wife to get in before we drove off (she had been outside helping me with the manoeuvre), our car was hit by an adjacent car, which was just starting to reverse out of a similarly tight spot. The driver obviously assumed we were going to head off and didn't know I was still waiting for my wife to strap herself in. Her rear bumper hit the side bumper of my wife's car.
I am a named driver on her car insurance, but she is not a named driver on my car, as she never drives my car. The damage to my wife's car is not too bad, so we are not planning to make a claim. If the other driver who drove into our car wants to proceed with a claim with her insurance company, whose insurance should I provide for the other driver? As I am a named driver on my wife's insurance (comprehensive), should I provide the other driver with my wife's insurance? If I did, and we are seen to be equally at fault, wouldn't both my wife and I suffer the consequence of the claim? Whereas, if I was to provide my own car insurance (although I wasn't driving my car), I will obviously not be able to claim damages for my wife's car (which we won't) and I will be the only one who will have a claim (whether it is a no fault or otherwise) against my insurance only.
Whose insurance should I provide the other driver?
Please advice.