Personally I don't share most posters' optimism. In my experience Insurance companies are notorious for trying every trick in the book to wriggle out of paying up, and when they inspect the vehicle they may well ask for MOT evidence.
However I don't think they will be able to welsh on paying for a courtesy car on these grounds, but they may reduce the size of the payout for replacement. It's not just a case as to whether the car was "legal" which it clearly was, it's also a case of what the woman's insurance policy small print says. However I'm not sure what they can legally enforce in the case of a claim on the Third Party aspect of the policy.
But anyway if they don't pay up for the courtesy car, you can always take out a personal claim against the woman, it being her problem that her policy didn't cover this aspect.
So although I may have muddied the waters a bit, I still think you can retain your peace of mind.