OK so I saw my dentist today to work out a "plan"... and in fact, they signed me off from their practice. They don't have a periodontist there, they apparently can't refer me to a private one (which I couldn't afford anyway), and if they refer me to an NHS one at Guy's it will take a year. "It's very advanced, you need to see a periodontist immediately, and there's nothing we can do until that's done." Yeah thanks.
So I'm now without a dentist at all, in a lot of pain, still very light-headed and on the verge of passing out, I can feel the problems spreading into the other half of my mouth, and my current course of antibiotics finishes on Wednesday morning. I'm pretty terrified about what will happen when I've had the last of them. Not sure what to do now. "Look on the internet", the dentist suggested helpfully.