As for the BBC it's firmly establishment leaning rather than left leaning in my view, and they also carry a lot of the blame for promoting Farage's poltics e.g. more appearances on question time than anyone else
As much as I hate everything that is Farage & co, I think that is rather a symptom of the BBC being less biased than other outlets. If you want to claim to be unbiased, you have to give both sides of any argument a voice in a place like the UK, regardless of how terrible or great one side is.
"Left leaning" is of course relative. Some claim that Tony Blair is right wing, and Corbyn is the "middle ground". If you claim that then sure Britian is overall right-of-corbyn.
But if you take the standard "Lab/Lib/SNP/Green" is left leaning, and "Tory/UKIP/BNP" is right leaning, then as the left have have a majority every election it's quite reasonable to believe what the facts say.
Take one look at the election results and you can see there is a constant majority that's "left"
A lot of the complaints stems from a perceived view that UKIP were overrepresented on Question Time in the run up to 2015. That's no defence for the claim the BBC is "pro establishment", as Farage and indeed Brexit was quite clearly anti establishment.
Yes, he might not be old money but he's as establishment as they come - look at his clothes he even wears the uniform of the establishment
One of his greatest tricks is that we was able to convince so many people that he was a normal person just like them