I am a motorist and I pay insurance, a lot of insurance. Some of that insurance money is used by insurance underwriters to cover the uninsured losses when the liability lies with an uninsured driver. That must be a significant amount to cover the size of payout required when an unisured driver causes an incident.
Surely a tiny proportion of this money could be used collectively by insurance companies to provide a group third party cover for any non motorised vehicle? Maybe it could be considered part of the insurance tax that come with all types of insurance premium, not just motoring.
This would then allow a proper amount of cover for any injury, loss of life or damage to property caused by any person on a non motorised vehicle used in accordance with the law on the public highway.
It would mean that there is no individual policy or premium for the cyclist, nor any paper work other that which follows the incident. If the cyclist was legally allowed to be cycling there and caused an incident then the insurers will pay out. If the cyclist was not legally allowed to be cycling there then the injured party can make a personal claim against the cyclist with the law on their side.
This will allow all road users to be covered either by payment for a motor vehicle or by non payment group cover if they are on a bike, skates, skate board, scooter, etc., regardless of age or means.
If a cyclist wants any further insurance cover, say theft, then they can get that separately.