The UK is at the forefront of the GC/TRA divide. You have three main parties here and two smaller parties (and also a bunch of very small ones that won't get MPs elected as they don't have enough voters). This is a brief summery...

The Conservatives (the right-leaning party. Historically for the upper and upper-middle class people. In power since 2009 and for the forseeable future. Their politicans lean more towards the GC side and are more vocal in speaking about it. The Covid respose has gone so badly though that they are spooked at making themselves even more unpopular by clarifying or changing laws.

Labour (Left-leaning party. Historically for the working class but in the last election swathes of working class Labour voters switched to the Tories over Brexit and because they were sick of being taken for granted as Labour voters.) Labour are 100% full TRA on the face of things. They put out a statement saying that any women who hold gender critical views need to be 're-educated' and, if re-education doesn't work, they need to be 'expelled' from the party. The ironic thing is that the vast majority of GC women are former Labour voters but Labour have decided to listen to a very loud twitter mob and assume that it represents the majority of the country. They also do this on other political issues, constantly misunderstanding what their core voters actually want from them and keeping themselves out of power. You'll get some GC labour politicans but the backlash they face is huge.

The Scottish National Party - only applicable in Scotland but a very powerful political force. Lean towards the left. Want another Scottish independance referendum and that is their main purpose of existing. Full TWAW and are pushing through laws that redefine what 'women' means and anti hate crime bills that are so severe JK Rowling's previous tweets would count as breaking the law and could be worth prison time.

The Liberal Democrats - on the left. Got into a coalition government back in 2009 with the Tories and, still, nobody trusts them as far as they can throw them. Some people are starting to forgive them. Also full TWAW - wants to remove any medical requirements for transition and their previous leader (who quit after losing her seat in December) said that she doesn't believe everyone is born male or female. Believes that transwomen should be in women's prisons as they are some of the most vulnerable people in society.

The Greens - on the left. The most TRA. Pledged to allow children to start medically transitioning as young as they wanted to as it was against their human rights to wait until they were adults.

Most people in the UK don't know what is going on. They hear the 'statistics' about trans suicides and assume that 99% of transwomen are effeminate gay men who want full surgery. There are several grass roots movements such as Womens Place UK (who the Labour party deemed a 'hate group'), various ones about sports and schools and, of course, the LGB Alliance. There has also been a change in reporting where openly GC articles are starting to get reported.

TL;DR - There are lots of noisy people on both sides but the majority of the country still have zero idea what is going on.