Fwiw surveys of the UK and political beliefs show that the positions of the main political parties in this country are not that closely aligned with the public.
Socially we tend to be a conservative country but this has a couple of anomalies - as I say there's not much of a debate on abortion and gay rights generally have crazy high levels of public support (above 80%) which goes a long way to explaining why the T decided to tag along. There's a tendency towards conformity but we also very much value eccentricity. Just not in our neighborhood! There a whole bunch of ultra straight individuals who do all the right things at the right time - go to school, get a job, get married, buy a house, have kids etc who really struggle with anyone who deviates from this. You don't boast about any achievement. That's seen as crass. We find Americans who do this really awful. We don't like the flag waving 'we are the best in the worldness'. You don't 'get ideas above your station' in the uk. Anyone seen to, becomes a target. But at the same time we love an underdog. That's why we like the eccentric - the eccentric is who we would like to be but are too bothered by what everyone will think to actually do it! So you get people simultaneously worshipping the eccentric, cheering them on whilst trying to also undermine them and trying to bring them down the second they achieve because they've not followed the rules and have got ideas above them! (See the whole concept of 'selling out')
If you want to understand this in a microcosm we have the traditions of buying a pint in the pub. Everyone is equal at the bar. Everyone buys a round of drinks for the group in turn regardless of status. People who are flash and insist on always buying the round are thought of as show off and insulting everyone else. There's a certain pride in paying your way. Conversely the guy who doesn't pay for a round is thought of as tight and shirking responsibility. A working class lad will offer to pay for a round for an upper class gentleman. And vice versa. Its an act of both politeness and demonstrating you can pay your way and equality.
Economically we are a socialist country but we are cheapskates and don't like parting with our cash. This makes us less socialist than most other European countries. There's an underlying belief that goes back to the Victorian working class and some of the non conformist prodestant thinking (Methodists) about making good through hard work rather than taking hand outs. This makes us a halfway house between Europe and America.
We are firm believers in universal health care. Support for the NHS is a nationalist pass time. It's probably the thing that politically unites us most. A couple of political parties would like to dismantle in, in some way but honestly you'd struggle to find something more unpopular with the public.
And then there's politeness. A whole sub-system of communication which no one but the British understands the rules to. Don't rock the boat.
We are also weirdly fiercely anti-authoritarian and this is helped a lot by the way the newspapers, justice system and political system developed (three pillars of democracy).
Orwell was British for a reason though. He understood the dangers of the political left in the context of the British political leaning and how this left us vulnerable to a potential authoritarian left. But this is well ingrained in enough people to understand it.
We also have the experience of the wars on the UK. They left deep scars. A mistrust of Europe and at times something of a feeling of being let down by America. There's the national pride of 'having won the but also the fact we have so many who experienced the hardships of the wars and it's lasting legacies psychological. Our voting system changed after WWI - there was a sense of the betrayal of the working classes as canon fodder by authority and women had worked to fill the gaps left by men in war. Our love for the NHS was born out of it being something the working classes had 'earned' for serving their country. So the dynamics between the upper classes and the working classes shifted. Our love of rights grew from this rather than the American civil rights movement. In this respect we have developed very differently too. Racism is different in the UK and is connected to patterns of immigration / colonies rather than slavery and is linked to issues of British class division. There are stories from WWII of how the British treated African Americans GIs really well and much more equally and the British authorities were told they had to take measures to stop this so that they didn't go back to America and expect the same which really changes a lot of how you think about how you view the war and post war period.
The church has a lot less influence too - if anything it's more or less irrelevant now.
Finally you have to understand British nationalism and sense of self. We like to be the awkward squad. We are not like these foreigners in Europe (who culturally we are actually most likely) not are we like those bloody loud Americans who think they can tell us what to do. As a nation we have an over inflated sense of self but we also are big enough to actually have some influence too. We are also small enough for grass roots ideas (this appears in all cultural fields - from music, the arts, politics etc) to grow and take root at pace. Our centralised nature in London often helps this. We see ourselves as a leader not a follower.
This is why gender criticalness in the UK touches so many different points. Geographically it's easier for us to organise on a grassroots level. We have this long tradition of leading the way (or at least thinking we should). We don't like Americans telling us what we should think. We don't like 'crazy christians'. We have this socialist roots and political leanings that simply don't exist in the US and Americans don't grasp (the OP is actually a good example - we'd probably ask if someone was conservative or socialist rather than liberal).
This is why grassroots have been able to better challenge government because it can organise more effectively. Our legal system isn't political so again we can go against party lines in a much easier way. This means authority has less control. Our laws protect rights in a slightly different way and there's much more transparency across the board. We can gain traction faster because we are a smaller country. We don't have the religious element to contend with. The vast majority of individuals who have taken the lead on the subject in the UK have come from left wing or liberal backgrounds and this is fundamentally different to the US.
It's a really big and complex subject, but one worth exploring because of all the stupid (American lead) tropes about being gender critical is being right way.
It simply isn't true and shows a huge lack of understanding of UK politics and culture the emergence and development of gender critical politics within the UK.
I could write a lot more on the subject but I think this is a fairly decent (and long enough) reflection on the key differences.