Sorry, I can't give you good news, just my experience having had various types of car insurance cover on and off since 1977 and a few claims.
Your protected No Claims Discount will be protected but of course this will count as one claim against however many in what space of time your Protected NCD allows, some allow more claims and/or in a shorter time than others. Protected only if non-fault is new to me, sounds a cut-price half cover to me, with a 220 PS (or any PS level) car I'd want full cover but each to their own.
How long things take may depend on lots of things.
You have been involved in an incident no matter who's fault and no matter if you claim or not (such as parked up and you're not in the car and someone or something damages your car) so this is proof of risk, you have claimed so yes unfortunately your premium will probably increase at next renewal.
If you paid for, or your insurance includes legal cover then you can pass all this over to them to try to sort, different policies have deferent cover and different providers are different quality. If the accident can be proven or taken as third parties' fault (if insured) then your insurance and legal cover will get you your vehicle repaired and possibly your excess and other from third parties insurance - but only the car repair is guaranteed.
I learnt when I was young that low cost insurance is great provided you never make a claim, then you need good quality insurance which can sometimes (not always) cost more in premium but a lot less in time and hassle for the insured, things were a lot easier way back then though.