While there is a recent piece of legislation which means that all VEHICLES on the road have to be covered by an insurance policy, the over-riding rule is the one which (under earlier, but still current, legislation) states that every DRIVER has to be covered by insurance.
That's because you can't sue a car, only a person. If a driver causes an accident which leaves another driver requiring life-long medical attention, the injured person has the right to sue the negligent driver for several million pounds. The law states that the negligent driver must have insurance cover in place to ensure that (whatever his own financial situation is) the injured driver will still receive their payment.
So ALL drivers (learners, newly-qualified or otherwise) need to be covered by an insurance policy which can pay out millions of pounds if they cause a nasty accident. As has been stated above, driving schools have policies which cover their students, and some people are covered (as a named 'additional driver) on another person's policy. But, as long as they're driving a vehicle, they must be covered by insurance, however that is achieved.
Chris