If I did have a screw loose - well, of course, I don't know what I'd do.
It's *always* hoped that two relatives who don't get along will try not to take it out on anyone else. But (sigh) that doesn't always happen. Perhaps, if I were your mother and you and I were at odds, I might think that anything I sent might not reach the grandchild. I'm trying to think of a best-case scenario here on behalf or your mother, since you asked me to put myself in her place and I don't want to be TOO evil. ;^)
How your daughter will react to this sort of non-recognition really depends on you and your husband. If you are angry about it, she'll learn that it's something to be angry about. If you can get a more objective outlook so you can say, "It's too bad Grandma chooses to do/not do this, but we're going to love her anyhow," then she'll learn to respond that way. It's really up to you. You can take advantage of the sad situation to teach your children to be gracious to others who are not gracious to them.
If it'll make you feel better, my MIL (who has not always been like this but might indeed have a screw loose now) has voted most of this family off the island, so to speak. She is angry at most of us (I've been accused of stealing thousands of dollars - I wonder where I put it!) and has not recognized Christmas or most birthdays for a while now. She may or may not still be recognizing her great-grandchildren. It happens. I hope I won't become that kind of person some time down the road.
(Just read your P.S., and I would have supervised, too. I was wondering if alcohol had something to do with this, but you hadn't said, so I didn't want to assume. I would think she is blaming you and thinking the worst of you, since alcoholics do tend to be blame-shifting.)