I mostly agree with Phillip's answer, but I want to get more in depth with how woke is used and why it's hard to give a good alternative.
The problem is that while woke was once a real term created and used by the left, it's not usually used in the left any more. Frankly, I don't think it ever got huge traction outside of specific circles that already were heavily focused on discussion and debate on such topics. At no point in time was a random democratic voter likely to say "I'm woke!" However, even those who used to use the term have backed off of it, mostly because of how it's been translated in meaning once conservatives started using it.
What it was
So first we have to look at the original definition:
a broader awareness of social inequalities such as sexism, and has also been used as shorthand for American Left ideas involving identity politics and social justice, such as the notion of white privilege and slavery reparations for African Americans.
There isn't a single good alternative term I can think of that describes this concept perfectly. If you look at the part about sexual inequalities the term feminist fits quite nicely. However feminism is only focused on issues related to sex - and often gender, but let's not get into how those are different. Woke includes more then that. It is true many feminists are also woke when it comes to areas outside of sex/gender, but it's not required by the definition of feminism.
For the racial part, this is very similar to critical race theory, but since that term is also in the process of being misrepresented and turned into a snarl word, as I'll talk about lower, it's probably not a good 'safe' alternative. Anti-racism isn't a great alternative, but it works good enough.
Socially aware or social activist might better encompass all aspects that woke is trying to cover, but those are both very general terms and how closely they match up with the original definition of woke really depends on who you ask and how they define the terms.
I could go on trying to give you alternatives, but in a way it's pointless because what you are asking is not for an alternative for how to refer to what woke originally meant, but what it means when conservatives use it today...
Woke as snarl word
As it is now the term woke is used largely by conservatives, not liberals, and it is used not to refer to the original meaning but as a snarl word. Snarl words—which are often phrases not just words, but we still call them snarl words for some reason—are very general terms meant to indicate the group you're referring to is one your suppose to hate.
These words usually had a specific meaning at one time, some still do have a legitimate and still in use meaning, but when they are used as a snarl word they really aren't about the actual definition or how exactly someone fits into it; they're catch-all terms that are meant to suggest you should hate this group. So, phrases like PC culture, Social Justice Warrior (SJW), Welfare queen, nazi, mansplaining, gun nut, cult, Bible thumper and many many others can or are often used by at least one group as a snarl word to refer to another.
These snarl words do somewhat refer to a group, for instance, a social justice warrior need not be a welfare queen; I could call a rich liberal person donating money to support some minority base scholarship program a SJW, but not a welfare queen. However, the definition of what is or isn't part of that group are very unclear to make it easier to toss the snarl word label on whoever you want other's to hate. So for instance I could arguable call anyone who criticizes anything I say a SJW, I could call anyone who gets any government assistance, no matter how little or rather they arguably earned it, a welfare queen. I could call anyone who likes, uses, or owns a gun, no matter how reasonable they are in their use or about legislation about them, a gun nut. You can argue the terms are supposed to refer to more heinous examples then the ones I listed, but when someone is painting the narrative to demonize a group they can do a great job of convincing others even relatively minor offenses warrant a snarl word label, and they often do. The point is snarl words tend to intentionally be undefined and unclear to make it easier to apply them to whoever you want to tell others to hate.
Let me be absolutely clear; snarl words are used by both sides of the political spectrum, and by plenty of other groups that don't fit in an easy left vs right division. Human nature is to create groups and factions and then tell everyone that is part of their group that anyone part of the other groups are somehow wrong or evil. It's been happening for as long as we had recorded history and likely far before that. Snarl words are useful for this, and so everyone uses them. I wish they didn't, that humanity didn't have to hate anyone they saw as not 'one of us', but well I can't change human nature.
While there may be some conservatives out there who understand the literal definition of woke and are making real criticisms about the actual meaning and interpretation, by and large when you see a conservative complain about a person or group being woke they are not using the original definition and instead using woke as a snarl word.
So what do conservatives mean by woke?
As I already explained, since it's being used as a snarl word, woke is very general and ill-defined. In fact, I'd argue the word woke is even less defined than most snarl words. It's so terribly ill-defined as to make any definition of 'who is woke' difficult to make.
However, in general woke tends to be used for 'anyone of the opposite political spectrum as us'. In effect, anyone who would call themselves a democratic, liberal, or progressive could be called woke. How much they do, or don't, fit the original literal definition of woke doesn't really matter; all that maters is that they fit close enough to 'not conservative' to allow throwing the snarl word at.
As such, pretty much any term for 'not conservative' could potentially be used in place of the word woke as it's used by most conservatives. This is why liberals have mostly given it up; it's been generalized and demonized so much that it doesn't have much meaning beyond 'bad liberal folks'. At this point, the only individuals I see actively calling themselves woke often are the ones who are using it semi-ironically to mean 'I'm one of those folks conservatives would hate'.
I will back what others have said, that 'progressive' or 'social progressive' is probably the closest you can get to answering your question, but I also stress it's not that great an answer. A progressive implies something more than what the original word woke meant; while it's plausible a very progressive person would likely believe they fit the original definition of woke, it is not required for one to be progressive. But how well progressive fits the original definition of the word doesn't matter, only how well it fits the snarl word and I'd say it a close enough fit. No word will be perfect since the very nature of being a snarl word is that it's too open-ended to clearly define who fits in.
What about cancel culture?
Well, cancel culture is also a snarl word, but at least it's a bit better defined of one.
As already said by Phillip, having things canceled you don't like is a technique used by both sides of the political spectrum, and it isn't new. When tv sitcoms first appeared it was considered scandalous and forbidden to show a toilet in a restroom, or a married couple sharing the same bed, and when Lucy from 'I Love Lucy' revealed she was pregnant, the episode was titled "Lucy Is Enceinte", because apparently saying the word pregnant was a terrible thing. One of the first (though not the first as some like to claim) onscreen interracial kiss, in Star Trek, only happened because both actors effectively refused to film any alternative to force a network, afraid it would get them cancelled to air the kiss. And let's not get into the many books that have been banned or even burned over the centuries.
However, while there is no doubt forcing things to be canceled if you disagree with them existed for a long time when a conservative says cancel culture, they are using it as a snarl word. It's meant to be "canceling when it's something I don't agree with". Or, in other words "when a woke person gets things cancelled, but not when a conservative does it" There are progressives out there that vehemently argue comprehensive sex education would better protect teens from the dangers of STI, unwanted pregnancies, and abortions and would argue that the reason kids don't get full sex education they 'deserve' is that it was 'canceled' by conservatives; but I doubt any conservative would call that cancel culture.
As such, in theory you could say again that progressive is the closest to the left equivalent for someone the term cancel culture is trying to refer to. Really though, I'm not sure most who identify as 'progressive' would be vocal enough to push for cancelation. You might want a word that more accurately expresses not just that they have progressive views, but that they consider themselves someone who is going to vocally act on those views, so perhaps something like 'social activist' would work better.
Though again it's not perfect, the very nature of snarl words makes it hard to put a good definition and thus alternative term to them.