The BBC is actually more impartial than you give it credit for. Or, alternatively, it's even *less* impartial, because you can (and do) hear similar rants about it from the left.
The real issue with the BBC is not that it is biased one way or another but that it misunderstands what "impartiality" means. Take any position on which there is not absolute consensus, but some sense in which the majority opinion tends. But it doesn't matter to the BBC, which always insists on presenting issues as either clear-cut or evenly split, and never anywhere in between. This is wrong, horribly misleading, and ends up making both sides given equal weight even if it's totally unjustifiable to do so.
In the "pro-British" debate, for instance, the BBC isn't anti-British but does make the "anti-British" side have more air time than it strictly deserves. This is still bias but not against one side or another, merely against the idea that anything can be different from entirely unsettled.