srw
It's a bit more complicated than that...
Not strictly true. IANAL (but I know something about insurance). If there is no fault, then there is no liability and hence no claim. If it is reallly an accident, then you can't claim against the insured. If it is the driver's fault, then (s)he is liable in law for the damages (s)he causes, and that is precisely what third-party insurance (compulsory* for drivers) covers.The insurers are liable for any damage their client causes, whether deliberate or not.
[edited to add}
If the driver is driving illegally without insurance, or is untraced, then the MIB steps in as pseudo-insurer.
*note for pedants - yes, yes, I know.