While we don't see Daphne's grandmother until the book's final chapter when she arrives with the Cutty Wren, we know enough about her before then to wish she'd just stayed home. As one of the Gentlemen of Last Resort says, "She is […] a mixture of the warrior queen Boadicea without the chariot, Catherine de' Medici without the poisoned rings, and Attila the Hun without his wonderful sense of fun" (1.48).
She also represents the worst of aristocratic polite society and colonial nastiness—like the idea that non-European people are inferior. Charming.
Everyone in this book believes in something. Daphne's grandmother? She believes in "people doing what she told them" (4.19). Wonderful. Luckily, this isn't her only belief. She also believes "that a conversation consisted of someone else listening to her talking, so even mild interruptions seemed to her to be some strange, puzzling inversion of the natural order of things, like pigs flying" (4.47).
You're probably getting the picture that this is a lady who likes control, and you'd be right. She tries to control her son's and granddaughter's lives too. She "wanted to make sure [Daphne] grew up to be a proper lady and not marry a godless scientist" (12.125). We can only imagine the fit she would have had if Daphne had married Mau. (We would have loved to have seen that.)
But! Just when you think you can't take any more, her son—who is now King of England—manages to put her in her place at the end of the book. (Well, if he can't, who could?) And boy, does he. He calls her a "sharp-tongued harridan with the manners of a fox" (15.237), which sure sounds good, even though it doesn't make a ton of sense.
One thing to take away from Daphne's grandmother is that this is the type of person Daphne could be. She represents the extreme of Western civilization—and is it any wonder that Daphne isn't too keen on it?