I broke a filling last year. Called the NHS dentist up the road and asked if they were still taking patients, and how quickly could they see me. I got an appointment that afternoon, he decided that it wasn't yet seriously damaged, so ground it down a bit and added some new filling. Was charged for a standard check-up and told to see the hygienist since I hadn't been near a dentist for about 8 years.
Similarly, a non-seriously chipped incisor earlier this year - sharp and catching on my tongue, but not painful. Phoned up, got an appointment the next day, he smoothed it down and declared it fine, only charged for a check-up appointment.
Logic dictates that he has other patients, but I've never actually seen one.
Anyway, the lesson is that it's a postcode lottery. In some parts of the UK NHS dentists are a myth and the private ones are hard to come by. Around here, they seem to be desperate for patients.
(Teeth are one of the things I got lucky with genetically. Never needed more than a couple of standard fillings, in spite of generally breaking all the "it'll rot your teeth" rules.)