It's kinda gone off topic here, but to put my opinion in here i doubt very much that the floor was level in the first place in all directions. Chances are it was flat, very different.
I show my customers this when we discuss 'level' floors, its a bit odd but bear with me. I get a cd case and hold it out, we can see case is flat, but it does not mean it is level, to find a perfectly level screed unless it is poured is very very rare no matter how good the screeder is, but it could be perfectly flat, massive difference.
So if the guy laid UFH and didn't self level it then he didn't do the job as any experienced tiler should, chances are the adhesive hasn't been mechanically keyed into the concrete as the was a mat in the way, so this could ultimately fail anyway.
As for the shop saying there should be zero lipping with slate, twaddle. Slate is a sedimentary rock combined of layers of shale and
mud, so unless the sea was perfectly calm for several million years or God was feeling generous that day there is no way it can be flat, fact. You can't give slate a grade other than, this batch seems denser and holds itself together well.
It does look like he didn't go through and grade the slate, which if they have been diamond ground he shouldn't really need too, as they would be plained to the same thickness.
So where is this going, well who can say.
I think the shop have been at fault for saying that there will be no lippage.
I think the customers like a natural finish, but didn't anticipate just how 'natural' slate can be.
And i think the tiler is at fault as well for not talking to the customer and explaining what slate is like, also getting the job done too quickly and not levelling or paying attention to the possible lipping.
All in all i don't see how anyone will come out of this smiling. The customer isn't happy and once you're not then it's very hard to rectify. As for the shop and the tiler we don't know who said what, so it's a very tough call i'm afraid.