maybe get a vpn for apple tv. might also have to set your location with all apple devices for uk

There’s the BBC iPlayer app for Apple TV in the UK

The BBC is funded by a licence fee that is paid by people in the UK, so it’s only available within the UK.

UK only and it’s not very good

Probably not helpful, but you can get BBC America content in the AMC+ app on AppleTV. Or you can use BritBox.