Another work-around.
Go into your echo device settings, and and device options, and pick a non-UK language. For example, English (Australia/New Zealand).
Once the new language has downloaded and your device has switched, you will find that BBC on TuneIn works again.
Note:
if you just say "Alexa, play Radio 4", it will still default to BBC Sounds.
But if you say "Alexa, play Radio 4 on TuneIn", it will play on TuneIn.
If you say "Alexa, wake me up to Radio 4 on TuneIn", it will ask you what time, and then wake you up with Radio 4.
If you say "Alexa, play Radio 4 on <name of multi-group>", it will play on the whole group, even though only one device has been changed.
Obviously, this is not ideal, as it may affect other country-specific skills that you use.
But I currently have an English (Canada) Dot running in my lounge, as well as my other devices - normally muted, but available for setting up multi-room playing.