It used to be the case that the super rich didn't need car insurance if they could prove they had ring fenced a minimum amount of money to meet any claims, I think it was over £1m, by making a "specified deposit" with the Accountant General of the Supreme Court, against their liability for injuries to others (including passengers) and for damage to other persons' property resulting from use of a vehicle on a public road or in other public places.
When the law changed recently and continuous insurance enforcement was put in to effect then all cars that were driven on public road needed third party insurance as minimum.
I don't know if the senior royals ring fenced money instead of buying insurance.