The radio channels are carried as audio only channels on basic digital satellite and terrestrial, you don't need a "smart" TV. They're all at the back end of the EPG. I think Sky Digital had them from the start, but On/ITV Digital DTT might not have, I think they were added when Freeview launched. Sky used to have the Music Choice service this article mentions as well IIRC.

I think back in the day various BBC Radio channels were alternate sound channels on UK Gold (which BBC Worldwide had a stake in) on analogue Sky, but I might be wrong about that.

Annoyingly, BBC Sounds app isn't available on Apple TV boxes, although I can just use AirPlay I guess...