Since the family is planning to renovate the old bathroom, it would be a good idea to install underfloor heating (just in the bathroom). Doing away with radiators should free up space, especially on the wall.
I'm assuming the existing screed will have to be milled – or would it be better to remove it from the entire room? What will I have to pay attention to?
Local conditions!!! Will all the sanitation items stay in their original places? If not, you may need to screed anyway. You should also pay attention to the condition. The disadvantage of milling is the lack of optimal thermal insulation at the bottom and the uneven heating of the tiles at the top. In a 5m² bathroom this might not be too bad, but in a 20m² bathroom you have to make sure that the heating load is sufficient. Also, grinding always produces a lot of dust and dirt that is so fine that it gets everywhere. So it may well be that chiseling produces less dirt than milling (extraction notwithstanding). Honestly, I can't judge the price.