It is of no concern to the cyclist whether the driver has a licence or not - that's up to the police or her insurer.
The OP seems to have focused on her licence, perhaps that's more of a thing in Australia which, given the OP's screen name, he is presumably from.
What the cyclist requires as a minimum is the registration mark of the vehicle, and preferably the name of the insurance company.
I try to carry a pen and a scrap of paper in my trunk bag, even if it's only the Greggs bag for my sausage roll, on which to write such basic details.
If the roadside exchange proceeds in a gentlemanly manner, there should be an exchange of details, thus it is also reasonable for the driver to ask for the cyclist's name and address.
@vickster has a point about being a woman living alone, to which there is no easy answer.
Were she to hit me in my car, I would like to think my conduct at the roadside would at least give her the confidence to provide me with a name and email address or phone number.
Going back to the OP, I agree he doesn't have much of a claim, so having reported the matter to the police, his best bet is to hope he hears no more about it.