I would say it was their fault.
But before you go cracking up like I would normally do, think of what their answers are going to be.
My guess is they will say wear and tear and nothing to do with what they done. And in a way they might be right. I dont think you would have any comeback on it.
But if they have filed it down and it looks bad you might have a case.
And did they really file the nut down or did they replace it with a new one? if replaced with a new one it might not have been worn down a bit like old ones get and your new string might have been putting too much pressure on it?
If I were you, I would buy a new nut and a set of very small fine files and do it yourself. It is not that hard to do.
another thing you could do is take it back with your old strings on, dont tell them you changed the strings and just blame them. But that would be dishonest,lol
ye get some that are cut out for the job and others just get by from pretending