The traditional "british teeth" was the UK's dental industry focussing on healthy rather than pretty.
Nowadays, it's caused by underfunded patient slots at dentists.

You can find a private dentist pretty easily, but it's quite hard to get taken on as an NHS patient (which means when you need treatment for something, you're not in the capped NHS bands). Which is especially bad if you're eligible for completely free treatment, as you're blocked by available dentists.

The dentists are generally given funding (or access to funding) for a set amount of NHS patients to make up the difference between NHS capped costs and their true costs. And unfortunately, there often aren't enough slots.
I was lucky with my current dentist that they happened to have slots when I signed up. And a few years later, they let me know when slots were opening so I could add the rest of the household.

Teeth are a luxury bone, you don't need them to work.

The UK having such horrendous teeth is a myth

source

I mean it doesn't well it didn't anyway until about a decade ago. You used to be able to get a dentist easily enough then austerity happened and look at us now! World leaders in shooting ourselves in the foot.

The British smile is really only a thing because teeth straightening and whitening aren't usually covered by the NHS and nobody cared enough to go private, everyone else has a crooked smile anyway. Your more likely to get bullied for braces than having a tooth out of place.