Managed to get through to the emergency dentist this morning. The receptionist asked why I was trying to get an appointment through them instead of contacting my regular clinic. Apparently NHS dentists have a legal obligation to provide emergency appointments with 24 hours of one of their registered patients asking for one.
Except, when I phoned my clinic about my gum, the receptionist there refused to give me an appointment, because I already have one scheduled for April 3rd. She was interpreting my request as my wanting to move my appointment up a bit.
Extra b_s is that dentists don't tell you about this requirement of theirs. It's left unsaid, for people to find out by accident.
Luckily I managed to get in today. Apparently the key words are "emergency appointment". It was originally "come at 12:30 and she'll slip you in between other patients", but they phoned me back with a few minutes later to offer me a newly cancelled appointment slot at 14:30 instead, which worked out better. Why do dentists always insist on poking the bit you've just said was hurting with a metal pointy thing? Got a prescription for amoxicillin, which I filled on my way home, so it's all worked out in the end. Still got my appointment next week as well (I forgot the time and it was phone-booked so no appointment card, so gotta call tomorrow to find out, when their computers are hopefully working so they can check), and they're going to start my referral to a surgical clinic once their computer systems are running again.