See https://radiotoday.co.uk/2025/02/most-bbc-radio-stations-to-become-unavailable-for-international-users/.
Use a VPN, like NordVPN or similar. Select an IP in UK
The same advice was given where I first learned this, and I won't do this.
I assume this isn't legal...
The VPN itself isn't illegal; inhowfar it is immoral to use it to bypass geolocking is a personal decision. However, replying the "Have you paid your license fee" popup with "yes" when you haven't is at the least a violation of the BBC's terms of service.
Not a good idea, then.
You currently don't need a licence to listen to radio programmes, but maybe you have to reply "Yes" in order to gain access to them. I don't know. I have to say it all seems extraordinarily petty of the BBC, but, as I say, I don't know the ins and outs of this situation.
Hrm. I am outside Britain, and I am able to listen to the latest Composer of the Week on my browser - apparently one can do this without signing in to one's BBC account, though it still knows of course that one's not in the UK just from one's IP address as one clicks play... seems not to have an objection (yet?). Less convenient than the app (not necessarily by much, though not logging in means I can't save my space automatically), --- though frankly with some apps that convenience not always present equally in all things anyway, so...