they'd pay, but, if you are driving wthout due care and attention, indeed any criminal offence, can't the insurance company sue you if there are excessive public liability costs, issue has been discussed in earlier threads.
They will pay out third party, not necessarily for the policy holders own loss in cases of fraud basically.
Cases of dangerous driving are not valid reasons for voiding a policy, which is what your talking about.
The ins co pays the 3rd party claim (they are forced to) but due to extreme circumstances they will void the policy and then go after the (now ex) policy holder via the courts for their money they have been forced to pay.
Its really actually quite rare, whats far more likely is they will void the policy and the (now ex) owner should declare this on any quote for a new policy which will result in lots of declines from insurers, and probably no online quoting engine will actually return a premium, most will need some human intervention.