Okay, first of all, why do both mothers run away? This happens all the time in fan fiction -- it's amazing how many wizarding children are abandoned by one or both parents for no real reason, other than that the author only wants to write about one parent (or neither).
So if you are really going to go that route, don't just make it an incidental detail to explain why Dudley is raising his granddaughter. The fact that Dandelion's mother did the exact same thing her mother did is going to be very significant. Also, where is Dandelion's father in this? Are you going to conveniently write him out of the story, too?
Now, as to how Dudley would react -- unless you want Dudley to be the villain of the story (which it appears you do not), I don't think he'd try to "squash" Dandelion's magic the way his parents tried to squash Harry's. On the other hand, I think he would freak out, especially if his daughter never displayed any magic. He might have worried about that, that magic might run in his family, but when his daughter turned out to be a perfectly ordinary Muggle, he probably breathed a sigh of relief and figured he'd dodged that particular bullet. Then suddenly his granddaughter is making strange things happen. Dudley might try to ignore it or justify it at first. ("No, the pudding didn't really float across the room, I just moved it myself and forgot... hah hah, I must be getting old....") But, unless you really want to make him as thick-headed as his father, I don't think he'll be able to keep that up for long.
So then he realizes she's a witch. He's going to have a fit. He's going to worry that she's a "freak" and will never fit in to normal society. He's going to worry about how his parents will react. What will they do when they find out that their great-granddaughter is one of them? He may well decide that it's better to hide it from them.
Then he's going to remember how he bullied and tormented Harry all through their childhood, and all the chaos that magic caused in his life, and how there was this terrible Dark Lord who was going to kill them all. Is his granddaughter going to suck him back into that world?
At some point, he'll know he needs to talk to Harry. What happens then is going to depend a lot on what you imagine Dudley's relationship with Harry to be like at that point.