> I have certainly never thought of the BBC as biased
With respect, I'm not sure why your opinion is relevant to whether it is biased. There are certainly many instances of bias, that like Twitter, only appear to occur in one direction. The Emily Maitlis opening to Newsnight in May[1] is one of the most egregious examples.
For opinion on the innards of the BBC you can listen to Rod Liddle's (former editor of the Today programme) many, many stories about it, or John Humphries, or for instance, this interview with Robert Aitken[2]. He's promoting a book called "The Noble Liar: How and Why the BBC Distorts the News to Promote a Liberal Agenda", and whether or not you feel it's right or even right for him to write it, it surely hints that some scrutiny might be applied to claims of unbiased centrism.
If you're looking for BBC reporters who you should watch for a bit of bias, Dominic Casciani is one. As to the Guardian, I was a reader for 20 years but I had to stop because it's seriously lost its way. Any given day I can find outright lies, let alone bias. I could go on but this piece[3] by the Guardian on Humpries' claims of bias at the BBC includes several quotes about his Brexit bias. He voted to remain. Maybe they didn't know - aren't journalists supposed to check?
[1] https://www.youtube.com/watch?v=d88goijIsas
[2] https://www.youtube.com/watch?v=aucDmK5E4bU
[3] https://www.theguardian.com/media/2019/sep/21/john-humphrys-...
Edit: I obviously can't number properly. I promise it's not because of bias :)