First off the MIBI appoints particular insurance companies on a rotation basis to deal with possible non insured claims. They act as "insurer concenered"
In many instances however, the insurer who covered the vehicle involved in the collision, will generally deal with the cliam of the Third Party.
The insurer concerned is not providing the uninsured driver with an indemnity and they have recourse to seeking recovery for any payments made. I would make an asumption that your dads insurance company did deal deal with the claim but as insurer concerned.
Some questions I would ask
As your dad was the policyholder did they ever write to him and let him know what happend re settlement of claim.
Did the insurance company reduce your dads No Claims discount - charge higher premium ? Doing so may suggest that they treated you as an insured driver at the time
Mary - the Motor Policy document really will be the key to your case.
In particular you should look at the exclusions under the policy and definition of Insured .
If it is the case that the wording was open driving ,You could (Your sols) argue that the policy wording did not give a clear and unambiguous definition of what consitutes a full licence, thus the courts can apply the Contra Preferandum rule and rule against the Insurance company seeking recovery. The Insurance company could be made indemnify you and treat you as if you were an insured person.
This case really deals with Insurance Law and as such it would be preferable to deal with a solicitor who has dealings with same.
In my expereince if you are unsucessful the MIBI and Insurer concened usually are quite happy to receive small affortable payments per month and your sols can negotiate that.