Both sides of the political spectrum regularly complains about the BBC being biased towards the other side. It's the best evidence that whatever bias there is either is minor or short-lived.
In my eyes, the biggest problem with the BBC isn't bias, but that they're terrified of seeming too biased, and as a result they're wishy-washy in ways that ironically makes them seem biased.
E.g. the left will accuse them of being right wing because they don't write in more critical terms about the governments willingness to deal with the Saudis, while the right will accuse them of a left wing bias for not writing in more critical terms about Chavez, and so on. Similarly they're often accused of bias for not reporting something, or very carefully reporting something, that either side believe to be fact, but where there's not (yet) actual firm evidence. Of course, if done "against" just one side, this too would be actual bias, but the BBC appears to be pretty consistent about it.
So we're left with reporting that is rarely very biased, but often exceedingly unsatisfying for everyone.
I'm not saying there's no bias - I don't believe that would be possible. But I do think that the allegations of bias against the BBC tend to be wildly exaggerated.