I just tested this on my Echo and Echo Show.
Here's what happened.
Me: "Alexa, play BBC Radio 1."
Alexa: "BBC Radio 1 from JWLV's TuneIn."
Music starts to play, no problems.
Me: "Alexa, stop."
Music stops and all is silent.
From what I experienced, this is all working correctly as expected. I am in the US, using a US Amazon account. When Alexa confirmed that she is to play BBC Radio 1, she said from TuneIn Radio. I didn't tell Alexa what music service she should use.
One thought comes to mind for those who cannot get BBC to stop playing. Are you using a BBC skill or some other skill or service that plays music? Maybe the skill is broken and does not respond to the "stop" command. For me, I don't have any skills enabled for anything that relates to music. All music related functions that I have on my Echo devices are those that are built-in, such as Pandora, TuneIn, and IHeartRadio.