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.
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.