I'm so surprised the majority of posters doesn't think it's an issue.
I would be annoyed too if a carer was taking my child to a nail salon without checking in with me first.
First of all, it doesn't seem like the most kids friendly places of all. I understand not checking in with you if going to the park, the zoo, the museum, etc - but a nail salon?
And surely your DD would have told her grand Ma that she had never been, so the excuse about not knowing it was a first time I can't buy it.
Good luck OP it sounds like you need to be more assertive with your MiL and I know it's not easy.