Have you tried going through the BBC Sounds app? Found the Tune-in app got pretty flakey a while ago. BBC Sounds far more robust in my experience. 

The BBC Radio 5 Live and Radio 4 stations are both working fine here and I’m based in the U.K. too? The stations work either with Alexa, or the Sonos App (BBC Sounds).

Because the BBC withdrew its stations from TuneIn amd other such “consolidators”.  As others have suggested, you have to play via BBC Sounds.

Via BBC Sounds on Sonos you can also get to on demand BBC content.