As someone who is both gay and left-leaning, I will try to explain the Fairytale of New York debacle for you.
Now, I'll start off by saying that I myself am not offended by the song in of itself. It's one of those things I've grown up with, so the f-word used in it doesn't really correlate to me with direct homophobia. However, for a lot of gay people it does, especially the ones who were subject to school yard homophobic bullying, or abuse in the streets where that exact word would be used. It offends and upsets a lot of gay people.
If we assess what's actually happening, Radio 1 have decided to play the 1992 version that MacColl and the Pogues performed on Top of the Pops, which used different lyrics for two reasons. 1) MacColl didn't feel comfortable using the word, and 2) it was broadcast before the watershed so that word wasn't allowed. It's also worth noting that Radio 2 will still play the original, and Radio 6 Music DJs will have a choice of which one to play.
I've tried to find the 1992 version and I can't find it widely available anywhere, not on Spotify or iTunes, only in video form on YouTube. The one with the slur in it is still available to buy and stream almost everywhere. So when you factor in all the above, it's not even censorship as it's still available from so many places.
Now, you may be thinking 'f****t' isn't homophobic anyway. Well, we can consider the ways in which this word is used. 1) Mr Brain's pork balls, 2) old Irish slang word for 'lazy' (which is disputed) and 3) a common homophobic slur. None of the first two are hardly common parlance, and the third is not acceptable for anyone to say. You direct it at a colleague in a workplace, you'll at minimum receive disciplinary action. And even if you don't think this insult is a homophobic slur in the context of the song, it's a powerful word with some deeply upsetting connotations attached to it. The question is: why would you want to upset anyone over a Christmas song? Especially someone who's part of a minority who probably had that word shouted at them as they were being beaten up in the school playground?
It's hardly a word you would use often, and it's no skin off your nose if you don't ever say that word again. A few people have told me that when they see straight people singing that song, they take great glee in shouting that particular word, saying it louder and with more enthusiasm because that's the only time they're now allowed to say it. And when they're told they can't say it anymore because it's offensive, they get upset and angry, lashing out at anyone who disagrees. And I think that's the thing that gets me annoyed. It's when straight people are throwing the thoughts, feelings and opinions of gay people in the bin over a word that's only common use is as a homophobic slur. If a gay person says something is homophobic, it's your obligation to listen to them and take it on board. They might not always be right, but 99% of the time they are.
I hope that answers your question.