Are race drivers required to carry a insurance policy
in the event of a tragedy like this occurring?He's in
for a long and expensive rehabilitation
Insurance, yes. You can't race in any FIA-organised series in Europe without at least some third-party insurance, and in F4's case, it's in Article 10 of the FIA Sporting Regulations for that category. The trouble is that not all insurance is equal, and F1-level independent insurance (which is compulsory in that series and covers to at least £10 million of expenses, even for non-third-party problems) is way outside the budgets of most F4 drivers, except those born into vast fortunes.
British F4's insurance situation is regulated by MSA. It has the same insurance policy rules for all car racing at national level. You can have independent insurance if you wish, but this is not compulsory. The two groups of people who are recommended to have it are non-British drivers (due to limits on amounts of treatment the NHS will cover for foreigners, that vary according to current state of agreements) and any driver who is expecting to compete abroad (the MSA deal only covers some such situations, not all - for example, anything requiring an International licence is generally excluded unless it happens in the UK or a nation with a full reciprocal agreement such as the Republic of Ireland).
It does, however, require payment into, and where appropriate use of, its in-house third-party insurance (as well as contribution towards comprehensive insurance for marshalls, that is considerably better than the comprehensive element for drivers) as part of licence fees. For drivers and marshalls alike, it covers up to £67 million (you read that correctly) of costs in third-party accidents. So if Billy's amputations had been a result of, say, a badly-maintained track sending him flying or a recovery vehicle in the way, Billy and his family probably wouldn't have had to worry about his medical costs for the rest of his life. He could have billed the MSA's insurance for all his needs and they could focus on everything except money.
However, third-party insurance is no help here, as drivers cannot be third parties in relation to one another (that's Article 10.4 of Formula 4's regulations, and the FIA does this for all racing series to prevent motorsport disappearing in a cloud of inter-driver lawsuits). So at that point, the compulsory comprehensive level kicks in... ...but if using the automatic insurance provided by the MSA, all a double amputation gets you is £10,000 (£5,000 per leg), plus £100 a day for the hospital stay. Note that Billy is in a NHS hospital right now and will only be charged for things like the entertainment unit and phone calls, so the £100-a-day bit won't be fully paid out unless and until he transfers to a private hospital for rehabilitation therapy.
Apparently, the family priced up the cost of prostheses to allow Billy to return to competition and discovered the typical cost to be £260,000 (not including any rehabilitation therapy needed to be able to use the prostheses over and above that which the NHS will provide). Note that although the NHS will provide free prosthetics, these are selected on price rather than functionality and are geared towards minimum living standards (to the point where many opt for a wheelchair in preference). NHS prostheses will be no use for returning to racing competition, which Billy has already indicated is his aim (and which his family guessed even before he awoke from surgery enough to communicate).
I have no comment as to whether private specialist medical insurance would have been a good idea, but can see why the Mongers - and, at a guess, most other racing families at this level - might think "we're already paying for one lot of insurance, why buy a second?"
(Sorry for the large number of edits. I kept spotting stuff I wasn't happy with straight after pressing "Save Settings"...)