It's a combination of factors stemming from unique political and cultural differences between the US and UK. I am not from the UK but I've done a lot of research on this because it's important to recognize why this happened in order to prevent it from happening here in the US:
1. The UK is extremely racially homogeneous which meant that intersectional feminism never caught on. Most British feminists are still stuck in the 70s with regard to the idea that there exists some kind of universal womanly experience (aka peak White Feminism) which influences their belief that people assigned male at birth cannot be true women because they didn't have the same exact life experience as a cis woman
2. The UK has a strong tradition of "skepticism" towards things that are perceived as anti-science. This likely originated in the 90s as a reaction to the rise of the religious right in the US, with British liberals fearful the same could happen in the UK. As a result many UK intellectuals, feminists, and comedians became ardent defenders of the idea that science can explain everything in life, and strongly opposed to any effort to explain societal problems with "postmodern" thought like the idea that gender is a social construct and not a biological imperative. These same "skeptics" eventually evolved into TERFs in the same way that the American atheist movement became ground zero for the alt-right. It does, of course, go without saying that it is the consensus of the entire medical and scientific community in the Western world that trans people are normal, healthy individuals who are the gender they say they are. TERFs claim these doctors and scientists have been paid off, the same way climate scientists supposedly have
3. Many people in the UK are heavily influenced by the British tabloid media, which operates off salacious, inflammatory, and false stories about sexually "deviant" subjects. It's sort of like if 75% of the US got their news from Youtube channels that think Captain Marvel is a cultural Marxist plot or something
4. The UK has a complicated relationship with "pedo panic" and the idea that there exist massive secret underground cults of pedophiles finding new and inventive ways to abuse children. There is unfortunately an obvious truth to the idea that abuse of children goes unnoticed, but the UK is practically a QAnon country in terms of thinking pedos are lurking around every corner waiting to abuse your kids. "Pedo panic" has historically been used to attack LGBT people and others who do not conform to heteronormative society
5. UK feminism is heavily influenced by a particular strain of 80s feminism that was very misandrist and virulently anti-male. These movements consisted of groups like "lesbian supremacists" and "female separatists," basically the female equivalent of MGTOWs or MRAs in the modern US. The "man-hating feminist" trope originated with these people whose views were shouted down in the US but unfortunately caught on among British women in the UK. They are obsessed with the idea that men are trying to control them, destroy them, and infiltrate their spaces, which is why they are so viciously hateful towards trans women
6. There is an extremely popular UK parenting forum called Mumsnet that, for various reasons, became infested with transphobes within the last 5-10 years. The site is basically 4chan for middle-aged white women. They go online to look for tips about changing diapers and instead find dozens of threads about how trans women want to rape their kids, which causes them to become radicalized the same way that young white men in the US are radicalized by Youtube videos