It means that you are socially aware. It's a phrase used by social justice warriors to indicate those who are now aware of all the inequalities in society whether it is by race, gender, sexual orientation, economic status or any other social terminology and that they are now willing to do something about it. Mind you this is just my interpretation of how the word 'woke' is now used.
Merriam-Webster agrees with you. When exactly did the word acquire this meaning? I'm a bit out of touch with social trends in North America...
Anyhow, Doctor Who is no stranger to being "woke", as this article from Den of Geek attests:
https://www.denofgeek.com/uk/tv/doctor-who/62051/doctor-whos-long-history-of-political-and-social-consciousnessTo those examples I would also add:
--"Terror of the Vervoids" from Colin Baker's final season
--"Oxygen" and "Smile" from Capaldi's final season, and
--"Kill The Moon" and "In the Forest of The Night" from his first.
In fact, I would argue that you can see how the resolution of this season's "Kerblam!" sets up the kind of macabre capitalism-gone-mad we saw in "Oxygen".
(And who can forget the image of the Daleks raising their sucker arms from "The Dalek Invasion of Earth"?)
The main difference from past episodes dealing with hot topics in social justice is that while previous attempts were much more allegorical, this season was far more direct and ham-fisted, with the TARDIS crew being dropped directly into some rather unpleasant episodes from Earth history, two still in living memory. And in two of those stories, the sci-fi elements were pretty weak and ultimately unnecessary. Upon further reflection, "Rosa" in particular was badly undercut by having a villain who was not only largely impotent, he was a pretty shallow, garden-variety racist. For all his faults as showrunner, I don't see Moffat handling such a character in the same way--he would give us somebody who would screw with the timeline JUST BECAUSE HE COULD. But I've learned that most people don't like the idea of evil that exists for its own sake, with no obvious motivation.
Then there's the tendency to use the new sonic much like a tricorder in Star Trek--but again, this is a writing decision, not an actual issue with portrayal.
The second major issue is Jodie Whittaker's characterization of the Doctor. There are a fair number of people out there who seem to think she's trying too hard to channel Tennant and Smith in her performance, and, while there may be something to that (they are the same age group, after all), I would also say in her defence:
--she/the producers likely wanted to make her Doctor as different from Capaldi's more detached, occasionally imperious and abrasive Doctor as Patrick Troughton's #2 was from William Hartnell's #1
--it took Capaldi about a season and a half before he really owned the role, IMO. Despite this being his dream come true, he didn't settle in as fast as Tennant and Smith. (I think that if you were to ask him now whether he played three different Doctors, his answer would be "yes".) Why not give Whittaker the same latitude?
It is true, though, that she hasn't been consistent, in particular consistently strong. I would say "The Witchfinders" was the most egregious example of that, where she just kind of knuckled under when faced with the misogyny of the early 1600s. It's hard to picture River Song reacting the same way...but in other episodes, she's shown the same assertiveness as her male predecessors. Not as arrogant as Tennant or Capaldi, but not as disarming as Smith either.
Ultimately, I think that if Whittaker's time as the first female Doctor proves to be a failed experiment, it'll be because she wasn't given the material to succeed, not because she wasn't inherently up to the task. Give her the best writers from the Moffat years and I think she'd do extremely well.
cheers,
Gabe