Hi,
I believe if the other driver is a named driver on the policy, then the 3rd party insurer should pay out for the damages to your vehicle. This will mean the main named driver on the policy will lose their no claims bonuses etc.
However if the driver is not on this policy and they have no insurance at all, the police should then be involved, and the accident will go to the MIB (Motor Insurance Burea). If this happens, it is likely you will have to wait alot longer for you're money then you would if the driver was insured.
Hope this helps,
Abi