So, was the story that the BBC censored the Pogues' song also made up by the Daily Mail? I have a strong feeling some of those stories actually are true, you know.

Yes, there was a "ban" - which received lots of press coverage. And then the BBC removed it.

I think it is fair to assume that this was because it had achieved its desired effect getting that press coverage. This served to remind people that Radio One existed, and to ensure that there would be extra listeners tuning in, out of curiosity.