Firstly, you must understand that the genesis of feminism in Britain is one that was heavily segregated away from other matrices of oppression. Lacking an intersectional influence you see in other countries' feminist struggles resulted in a feminist philosophy that had calcified as being upper class, white, and deeply homophobic. Many of the thinkers that popularised TERF ideology cut their teeth (so to speak) either in lesbian exclusionary branches of feminism (that argued single sex spaces should also be segregated based on sexuality and lesbians should be placed in the same spaces as men for straight women's safety) or as "political lesbians" (who argued that sexuality is a choice one consciously made and that women could escape the patriarchy by choosing to be sapphic). This is the environment that spawned such intellectual power houses as Germaine Greer, who wrote passionate defences of wanking to images of prepubescent boys. Being toffs, they despise poor people but have a particularly deep hatred of sex workers, hence the popularity of SWERF ideologies in these circles.

Popular support of minority rights are almost intrinsically tied to the labour movement due to a long history of solidarity campaigns (such as the NUM/LGBT solidarity campaigns during the Thatcher regime). Both the old school aristocrats and the newer neoliberals therefore hate these minority groups with the same vigour they despise the unions they worked alongside.

Britain is ruled not by democracy but rather of a system of rule by the most inbred toff. Thus any intersectional critique of society must jump the hurdle of its activists not being rich enough for MPs to even consider them human. Therefore intersectional feminists were locked out of political influence while the academic, exclusionary feminists were selectively chosen to inform on all feminist issues.

This creates a condition where support of queer rights is split along a class divide more than a left/right one. And thus transphobic policies are supported bipartisanly within parliament.

secondly, we have had uncontested Tory rule since 2010. Tories are a special breed of right wing ideology in a couple of ways. Firstly, they are died in the wool aristocrats of varying intensity. Tories operate on the complete assumption that they have magic blood they've inherited that makes them superior to plebs like you and I, and have barely concealed contempt for us at best. Secondly, Tories are motivated by an intense desire for one of these at any given time: wealth, power, death, or sex. A decade and change of these fucks (and the coked up liberals of New Labour before then) prioritising personally accruing these has resulted in a country completely gutted and hollowed out of anything capable of maintaining a decent quality of life for the overwhelming majority of the population.

The Tory party's main tactic in managing a grip on power is to create a system of culture war scapegoats. When I was a little one, that was Polish immigrants, Muslims and homosexuals, and I vividly recall the moment that they pivoted overnight to blaming Romanians, Muslims, and transwomen (trans men don't exist in the perceptions of this cultural milleu). As the wheels spin off and last bits of tape that hold this country together come flying off, Tories have ramped the moral panics to eleven.

Thirdly, after losing ground on gay marriage, homophobic groups both foreign and domestic saw the relative political obscurity of transpeople as an a wedge issue they could try to use to push back against gay rights. Money was splashed around to astroturf campaigns against transpeople and homophobes within the political establishment gave these groups the audience within parliament that the exclusionary feminists already had.

Groups like LGB alliance, are Tufton street housed astroturfs that exist as an attempt to drive a wedge between transpeople and the rest of the queer community (and then bisexuals, and sho on ).

Finally, and most importantly. Trans rights are not actually as controversial as parliament and the ruling class owned media want to make out. The BBC and tabloids and the like, have been artificially promoting transphobic voices despite polling across the country being heavily tilted in favour of trans positive reform to, like, the GRA and stuff. Although this is an issue with a gendered split, with more women supporting trans rights than men. That's not to say there isn't queerphobia in the working class (I've experienced more than my fair share despite growing up in a place with an otherwise high support of queer rights), just that the working class is less bigoted towards queer people than the toffs trying to pretend they have have a mandate to oppress us.