I can't get Tune_in to play BBC stations either. Seems to want me to pay for Tune-In Live service.
I can play BBC stations through the MyTuner skill though. You'll have to experiment with the verbal command. It seems different than with Tune-In.
Listening to BBC Merseyside as I type this.
In general though, I love the Tune-In performance on Alexa. Use that to access CNN, MSNBC, public radio stations, etc. I tried linking the Tune-In account at first but that served no purpose so I disabled that.
I did enable the CNN skill, and that provides a menu when I say "play CNN" (rather then just playing CNN on Tune-In, as it did before). But it's simpler just to have it play on Tune-In, so I might disable the CNN skill too.
To clarify, I'm in the US. I notice that the BBC skill listed in mr_a's post above is not available to me. Likewise, I'm not sure what skills might be available to those of you in the UK. Good luck!
I wake up the Radio Kent I just say play "Radio Kent" and it plays from TuneIn great, if I say "play Classic Country 1630" nothing, but if I say "play Classic Country Radio 1630" it comes on straight away. Thinking of local radio just tried "play Radio Bristol" got an American Country Radio station, one I will remember from now on to pay Local Bristol I had to say "play BBC radio Bristol" I get the local BBC one. Because I listen to Radio Kent most mornings I think she know so I no longer have to say BBC now and again she says you can get the BBC app but for me TuneIn is the best and a good selling point for Alexa, I tried the dark side Google and out the box you cannot play any radio so that went straight back.