I loved Nemo, but I would not allow my DD to watch it and she is 2 yo just like the OP's daughter. The reason? Well, the opening scene is really scary!! Pixar may not be Disney, but what the heck is with these children's movies where THE MOTHER DIES!!!!!. I mean, is this really a necessary part of the story, that mama and all the little baby fishes except one DIE??!!! I well remember being stunned by Bambi (at 5 years of age) and other similar movies, and it just pisses me off. Kids that age shouldn't have to deal with something so traumatic as the loss of a mother....okay, rant over.
I think the bigger issue here is your mother and how she is not respecting your wishes, then using lies and other sneaky tactics to get around them. If you don't put your foot down, this will extend into all sorts of other things that are more serious to you than a movie.
I personally would stop in the minute it happens. Like Lilyka said, right then and there when your DD came running in with the movie, I would have taken it from DD, apologized to her that Grandma could not respect the rules of this house, given it back to Grandma, and told her that I specifically said "no movies" so please take this back and don't come back here if you have it on you. I'd do this right in front of DD and Grandma so that DD learns a little something about the rules in your home being respected, AND that it is Grandma's fault that she is so upset, not yours. And I would hope Grandma gets embarrassed/upset enough to clue in that this is serious to you. My biggest pet peeve is family members who refuse to acknowledge your wishes about parenting, and keep pushing the issue. Thank goodness I have not had to deal with this, but I sure as heck would nip it in the bud if it started.
I also would not give in and let my child watch a movie 15 times a day b/c they are having a tantrum. I agree with those who wonder why you "HAVE" to let her watch it that often? You are the parent, and if your rules are consistent then the kids will survive it...