It's tricky to have a discussion about what is and isn't "woke" because there isn't one agreed definition of the word.
Some seem to think any time a minority is featured in anything (eg a TV show) it's automatically woke. Even though there are lots of minorities in this country and, therefore, it would seem logical that said minorities feature.
If gay relationships are featured in, say, a TV soap, some people will say "why do they have to have gay storylines? More of the woke agenda at play". Yet if me and my same sex partner moved in next door to someone like that, it's unlikely they'd say: "this street is becoming so woke", because it's real life - so there's clearly no perceived agenda. It's then just accepted that a minority exists nearby.
I do understand the arguments about 'positive discrimination' - eg when someone's protected characteristic plays a factor in being selected for a role. Clearly someone should get a job based solely on merit. However, I don't always have an issue with it because historically minorities have been disadvantaged and, therefore, it seems fair to level the playing field. At least in some circumstances.
I don't really see the BBC as being woke, though. I know Doctor Who and Strictly have been used as examples, but I don't personally care if Doctor Who is a woman or black because it's a fictional character who has always regenerated into different guises. I can't really take issue with fiction. With Strictly, why shouldn't there be disabled or gay contestants? They're still very much a minority in similar proportions to the real world.
Speaking as a minority, it's nice to see yourself represented on a large show when historically that representation just wasn't there. If I'd see two men dancing on TV when I was growing up, it would have helped me accept my sexuality more than I did. Of course it takes more than one appearance on a TV show, but when I was a teenager there was literally no representation - so I felt pretty much invisible.
It's a complex subject, but it's nice to discuss it civilly. We've all got voices to add to the subject and we should all listen to and respect each other. Otherwise we'll never move forward.