That's a bit of an assumption, where's your proof of that?i would like to mention your asking some of the best tilers in the uk on appropriate finish, they didn't qoute, so there finish will be different along with there price. I would not be happy with this finish no one would. If you would like the best finish possible there's tilers on here that can do it for sure so you could have a look for a trusted advisor near you.
As far as lippage at the tap possibly could remove the tile and sit a new one in, its hard to gauge without inspecting to see if it's one tile high or low. The window cuts should be redone properly but can your tiler do that and if he can why not in the first place because they can be a lot better than that.
If you could have a look at some of his other work to see what it's like that should give you an idea of what you have paid for (maybe) I hope you get everything sorted like others on here will agree it's upsetting to see this workmanship.
Pedant mode: there (their) sorry pet hate