I can't speak to the U.K.'s politics, not being British myself and not following very much of it. I can offer my perspective as someone who lives in the United States.
I've always viewed social justice as a positive thing -- at least, conceptually. Which is why it's so disheartening to see the term "Social Justice Warrior" used as a pejorative. I observe a similar thing happening to usage of the word "woke." Its original intent was exactly what you've described...making the world aware of injustices, and solving those problems to make the lives of underserved people better.
At some point along the way, a bunch of elitists from academia went off-the-rails with misappropriation in their attempt to control the entire narrative. For example, from many of their perspectives, it isn't sufficient enough for a White person to examine one's own implicit biases or unconscious racism and then challenge oneself. They insist that we outright identify ourselves as "racist." Then, when a White person says they're trying to unlearn different forms of racism, that same academic elitist peanut gallery (and their like-minded trolls on the Internet) will squawk how White people are "fragile" for not wanting to be accused of racism in the first place.
Um...did I fall asleep, at some point? In what universe should anybody *WANT* to be racist? Isn't eliminating (or at least reducing) racism THE GOAL, here????
And I do apologize if it seems as though I'm centering White people in this reply. Obviously, when confronting SYSTEMIC forms of racism, the voices and perspectives of BIPOC need to be re-centered and lifted up. The problem is that people of color aren't a monolith. There is no single "standard-bearer" to serve as a universal "go-to" ambassador of racial justice (or wokeness, or whatever one wants to call it).
So, for White people who want to be effective allies to Black, Indigenous, Latino, APIDA, and MENAPA communities...we are often given contradictory messages by different sources who claim to be "woke" and who advocate for racial justice. And that's fine, to a point -- because, again, it just speaks to the diversity of perspectives within ANY community. But certain people (not "all" or even necessarily "most") who are proponents of "wokeness" -- or who consider themselves Critical Race Theorists -- are peddling some very specific bulwarks of ridiculous dogma that not all BIPOC even uniformly agree with.
And, a lot of would-be white allies just clam up and retreat into their shells, because they don't want to be lambasted as "politically-incorrect." I personally don't have that problem...if somebody wants to call me "racist" or accuse me of racism, they can go right ahead and do so -- but the onus is still on them to articulate what exactly I (hypothetically) did/said that was offensive.