(NZ lurker) The media publicity around KJK's visit was so biased that most people believed she was a nazi hate-speaker and had no idea what she was actually trying to do. Personally, Laurel Hubbard raised the absurdity of the issue and some conversation around it, and KJK's visit encouraged me to look more into what was actually happening around the world. And then I came across mumsnet and properly peaked! I have still never seen anything factual in the media about KJK - when she announced she wouldn't be returning, the media acted like they had won. We don't have many media options unlike the UK, what we do have seems captured.
NZ is very much a follower in most things - the discussions happening in the UK are not happening here yet, and most people really don't know what's going on elsewhere. One of the minor coalition parties had a campaign policy around transwomen not using women's toilets, separate sports categories, etc. The majority party and subsequently the media made it into a bit of a joke - like is that all he's got to worry about, cost of living, bla bla? I'm not sure if it got through the coalition negotiations - I haven't heard anything about it so am guessing not.
Oh and yes, I'd definitely agree with the previous poster that our new government is slightly right of centre. The majority party is pretty centrist, with the two minor parties in the coalition a bit further to the right.