Perhaps - though the comedy, drama, radio, documentaries etc are mostly still fairly "centre to centre left" - it's mostly just the news reporting.

It often seems like the "shift in society" has been manufactured, and that that the BBC have been part of this manufacture - and that shift is really just a change in boldness, volume and support - people are loud and proud about hating people over race or gender etc.

I feel a far larger part of the shift was when the Tories threatened the BBC with funding cuts, cancellation and "sharing the licence fee with poor impoverished media tycoons like Rupert Murdoch" unless they removed their "left wing bias" (actually central bias at the time) and starting promoting a "balanced" (actually right to far-right) viewpoint. This was accompanied by sacking and replacing key staff members with their own right-promoting appointees.

That's continued and got worse - they've spent much of the last decade promoting the voices of increasingly extreme right-wing viewpoints, and either silencing, mocking or generating controversy about any minor centre or left-wing viewpoints.

None of those people changed when the government did, so it's stayed in the right and possibly drifted even further. Remember that in terms of many of its social policies, our current Labour government is more right wing than Cameron's Tory government was.