You must be really shocked. Drunk drivers are scum.
Unfortunately the insurance process is likely to add insult to injury. It's likely to be expensive, to feel like a kick in the teeth, and some insurers will make your DH repeatedly describe the accident.
The following all applies if the third party insurer will not pay out. Even though you know the car registration, if the keeper denies being the accident driver their insurer will probably contest the claim.
Your insurance may include a courtesy car, however these are only provided until your car is repaired or written off. So you might not get one at all if your insurer is already confirming the writeoff.
When the car is written off you will get the market value of your car minus your excess. As a PP mentioned there is often negotiation here, insurers giving low initial offers which get increased. You can dispute the decision to write off, or depending on the type of write-off you can buy the car back.
If your car is written off your policy ends, but if you are paying monthly then in most cases you still need to pay. This is because paying monthly for car insurance is technically taking out a loan for the entire premium that you pay back over the year. And your new car will need a new insurance policy. If the old policy only recently started this can be a considerable extra expense.
Depending on your insurance, there may be no or limited coverage for any injuries DH sustained. Just because he seemed OK doesn't mean he's not been hurt, some injuries aren't obvious straight away. It depends on exact terms and whether your insurer tries to wriggle out of paying.
"Fault" to a car insurer isn't really about who caused an accident, it's about who pays. Therefore this claim goes down as at-fault for you, with a corresponding greater insurance premium in future. I wouldn't trust what your insurers have said; they could be assuming the other driver gets found and change their mind if it's not.
So yeah, a shit situation all round.
On the other hand if the drunk driver can be traced, and was insured, and their insurance pays up, things look somewhat better. No excess, no at-fault claim on record, a hire car may be available (but beware! these have pitfalls that can cost you £1000's!).
Either way you'll probably spend the next few years getting ambulance-chasing lawyer spam calls, just to add a final insult.