This a great place to learn about chinchillas. We have some great volunteer help as well.
The good news is that chinchillas, when they come from a reputable breeder and are given a proper diet and environment are relatively care-free. The bad news is they do have to see a specialist vet and are more fragile than rabbits. So when they do get sick it can become very serious quickly.
I've owned chins for over four years now. My first one came from a pet store and was essentially a rescue so it wasn't being properly cared for. It presented with malocclusion six months after I had him and took him to the vet to diagnose it. Then I had had him euthanized. I think it was less than $400 (USD) for that ordeal including a physical exam of the teeth under anesthesia. When malocclusion is diagnosed it's very very rarely worth trying to save a chin. Prevention is the best cure.
The second chin was from a reputable breeder and he presented with a coccidia infection that was easily treatable. That was about $150.00 maybe six months after I got him. A couple of years later he presented with a watery, crusty eye. I'm suspecting it was from getting dust or hay in it. As a precaution I had that examined and got medication to treat it. Turned out I didn't need the meds. That was about $90.00. I was being cautious on that because watery eyes can signify a respiratory infection which must be treated aggressively if it were.
My third chin (now I have two in the same cage although divided) had some unusually formed feces since I got him and I had him checked out at a different vet just to make sure everything was okay. After consulting with him I came to the conclusion that his intestines are formed just a little differently than usual and it was not a problem for him. That was a $100.00 visit. I had the option of getting a fecal culture for an additional charge if needed. The exam was good peace of mind though and he was found to be very healthy and they did a quick visual tooth exam as well.
So I can say only two visits were for treating or diagnosing a real problem. The other two were mostly precautions which amounts to about one every two years per chin. As they get older, in their teens, they may not do as well. If you're someone who is hands-on with care you can manage them as best you can until it's time to put them down if they get something more serious.
I would also mention that I've treated minor cuts and scrapes myself and I know what infection looks likes that requires veterinary care.
Chinchilla's digestive systems are more sensitive than rabbits. You can avoid a lot of problems by not feeding treats and make sure their diet is very plain. All they need is good alfafa-based pellets plus loose timothy hay. If there are digestive problems GI-stasis is pretty common. However it is rare if you catch it early, monitor their daily food intake, and syringe feed a recovery diet (at a veterinarian's advice if possible) to keep those intestines flowing. The problem comes if they stop eating for several days and it goes unnoticed. Also they are very sensitive to oral antibiotics and must be monitored intensively if you ever use baytril. By weighing your chinchillas once a week you can usually find warning signs well before it gets that bad.
It is not recommended to let chinchillas and rabbits together. One is they don't have the same behavior patterns and the potential that they will injure each other while playing is very high. Most importantly is they can infect each other with diseases. A bacteria that is normally present or beneficial in a chinchilla, for example, may wreak havoc in a rabbit, or vice versa. You certainly don't want your chins to have access to vegetables or rabbit food. (However certain specific types of rabbit food are suitable. Most are unacceptable to chins).
Exotic pet insurance is certainly possible. Considering how little care I've needed I personally wouldn't find it worth the expense. (Although I don't know what the UK insurance market looks like. I imagine it's cheaper). For me it's worked well just to have a line of medical credit that can be used for veterinary care. Basically I've set a limit on how much I'll spend an a vet before I have might have the animal put down.
And lastly handling chinchillas properly and preventing accidents always a good policy. Prevention is the best cure for a lot of what I've seen. Make sure your cage and accessories are safe and that handling them will not land them on their face or result in falling.