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.