Hi,
sorry for the late reply but I wanted to investigate this before answering. Turns out this has got nothing to do with the DNS server(s) used at all (I also tried this).
Unfortunately at least one of the IP addresses of the server London1 is recognized by the BBC as not being inside the UK, so I was able to reproduce the issue. But when using a VPN protocol (I did my tests using our Windows VPN Manager with OpenVPN) you can simply reconnect to the server (or use the other one in London) while having the "use random exit IP" option (located in the IPs and Ports tab in the client or on the settings page in the member area of our website) enabled, and you will most likely use another IP then, as both servers in London have six IPv4...