I see a lot of claims here, but chiefly:
- A 1990s "British skeptics movement" was directed against "postmodernism"
- Such skeptics are especially "no-nonsense" people (humorless?)
- Such skeptics are opposed to "individualism"
- "TERFs" are attempting to gatekeep Britishness
To an American all of this seems bizarre. But if we limit our claims to 1 through 3 I think I know what is being talked about here. I previously addressed this mysterious British conception of skepticism in my answer to "Is the skeptical movement an offshoot of the Communist Party?"
To summarize, in the 1990s there was a loose network of British libertarians that emerged from a Trotskyist cult called the Revolutionary Communist Party led by a professor named Frank Furedi. Critics of this network call it the "LM group" although it is not itself a cult -- for example, some people in this network were simply Furedi's PhD students and acquired their ideas from him in a normal academic way.
I'm not aware of an objective description of this network's influence on British intellectual life, if such a thing is possible, but there is a good article about them in the London Review of Books and you can kind of guess from this description how the network's prominence influenced British views of skepticism.
Furedi’s talk has another odd rhetorical habit, which I noticed was copied by other speakers at the [LM group's] Battle of Ideas. ‘You and I as grown-ups’, ‘not just as biologically mature grown-ups’, ‘the experience of grown-ups has become pretty irrelevant’: the IoI adores grown-ups, and being grown-up, and talks all the time about how important it is to treat each other ‘as grown-ups’. The effect is paradoxical, but predictable. If you talk constantly about ‘grown-ups’ it makes you sound like a child.
[...] The one thing, perhaps, that ultimately holds the LM network together is its members’ refusal to countenance the existence of psychic conflict or confusion. People are not ‘hapless, fragile victims’; neither do they struggle to contain ‘the beast within us all’. People run fine on ‘democracy, science, reason’. We are, after all, ‘grown-ups’.
The LM group is harshly critical of postmodernism, in a much more confrontational way than 1990s-2000s American skepticism which was more directed towards the religious right and New Age claims. The strange claim that skeptics oppose "individualism" can also probably be traced to Frank Furedi's critique of "the self-directing individualism advocated by therapy culture" found in his 2003 book Therapy Culture. This anti-psychiatry spread throughout the LM group. (It's a slightly weird use of the term "individualism" by the NYT author -- Furedi champions individualism in the sense of self-reliance.)
I hope this helps you understand the likely context of these strong claims about "the British Skepticism movement." I have no idea whether or not the two women named have unconscious influence from Furedi and the LM group, which is the seemingly unfalsifiable claim being made, and I doubt an objective answer can be provided.
I also don't know what the point is of this unprovable accusation. The libertarian Furedi/LM network was frequently contrarian, but they are not generally seen as a blot on British intellectual life, as far as I know. I don't think the kind of heated political arguments around "critical race theory" in America have emerged around skepticism or libertarianism in Britain. While the "no-nonsense" rhetoric can be seen as a bit condescending (as in the quotations above), I'm not aware that it's considered damaging to feminism, etc.
Finally, claim 4 is separate from this. It emerges from the so-called "TERF" contention found elsewhere in the NYT article that gender dysphoria "has been imported into the U.K. by America". Although it is stated as if it logically follows from claims 1 through 3, based on my knowledge of this British intellectual history, I don't think it is actually related to these claims which are also unproven and seemingly unprovable.