With lvt it will need screening before to make flat as mentioned above, so it’s just as easy to put some ply over the top of existing to level out (or screed if you have concrete as also mentioned above) to lay tiles. They can be cold though indeed. Have you thought about an actual lino?? Some cushioned commercial ones are incredibly durable and it would be warmer and much cheaper than lvt. I would personally either go with the same or complimentary tile as on your wall (love the idea of the hexagonal to match your wall) or a contrasting tone in a different effect if it doesn’t match exactly

Mirror was a Wayfair Bargain Sam - half price ! The niche and shelves should have a tiny weeny fall on them, practically imperceptible, but enough for the water to run off them. I don't know what colour your tiles are, but maybe you could try that non-slip matting stuff that you can put on the top of washing machines, counter tops to stop them getting scratched. It's got holes all over it, so the water will run off, but it will stop your bottles slipping. They sell it on Amazon and in Dunelm. Can't find it on Dunelm's website, but i know they have it in there. Here it is on Amazon, not sure what other colours you can get it in?

I would look into microcement so you can have the same surface all over - will make the space feel less cluttered, and no grout lines. You can get it in lots of colours. I’ve never had it myself (yet) but it sounds great. For your bath, you may find you need to chop out some plaster to get the bath in, and then re plaster around. I’ve done this before with a 1600 bath in a tiny bathroom. How much space do you have apart from the bath space? Is the door in the corner - if so then you only have 800 of wall to play with? If that’s the case it’s going to be very tight fitting a loo and sink in there, you might need to look at those loos with a sink on top of the cistern although they look awkward. Definitely re hang the door to open outwards!

If you're going to have a fitted bath (rather than a free-standing one) then it makes sense to put a shower over it - so you have a shower head to rinse the bath at the very least! Assuming it's a kids bathroom more than anyone else, then I think a shower over bath is perfectly acceptable in exchange for more space in the room. I'd put in double sinks though, so more than one kid can brush teeth at the same time (or preen themselves as they become teens) - also makes it feel a bit more fancy. There should also be room for decent storage too - just having more space in the room will make it feel more luxurious too.