nobody is handling a claim yet, neither of the parties dealing with the mother are insurance providers.

I mean "the process" rather than "the claim". If you have an accident you don't have to handle it with insurance, you can fix your own car privately if you choose to.

AAMI has been contacted as per the post, AAMI also doesnt need to be involved in a no fault accident, you may want to involve them but its not required

AAMI was contacted after the car was towed to a repair centre by another company. This is not a no fault accident, OP's mother hit another car and is at fault. Not involving them isn't required, the owner is free to pay for both cars to be fixed if they choose to.