I can see Dudley and Harry developing a low key relationship, where they simply stay in touch through cards and brief visits. If Dudley has matured and really grown through his self examination, he would probably leave his parents out of the mix. Let me reword that. Wisdom would dictate that they all leave his parents out of the mix. Petunia will be even more bitter now that her own son has decided to like his cousin. Vernon probably gets more blustery and irrelevant as time goes on. If Dudley knows what's good for him, he lives several hundred miles away from his parents, is dutiful, and keeps his own council.

Petunia and Vernon Dursley would definitely not be a part of the equation here. Vernon is and has always been a git. He has such a high opinion of himself that I doubt that his stay at the safe house that the OOTP fixed up for the Dursleys during the trio's adventures, had any affect on Vernon at all. I think that Petunia has frozen into her bitterness now, and can't change her behavior for the better. Maybe if Vernon's high blood pressure resulted in an untimely demise, she might, just might make an improvement in her attitude, and be willing to let go of her animosity towards Harry.

Well...I could be going out on a limb here, but perhaps part of Vernon and Petunia's treatment of Harry stemmed from a fear that Harry would "turn" Dudders into a wizard, himself. As we see with his encounter of the Dementors, Dudley was able to sense them on a lower level, so although he wasn't a full blown wizard, he was more sensitive to Magic than either of his parents were. Also thrown into the mix was that Petunia felt ostracized from her sister because of Lily's pronounced "magicalness" - would Dudley ostracize her in a similar manner if he was chosen for Hogwarts? It would just add insult to injury if Dudley turned out to be "worthy", and Petunia was not, even after her impassioned plea to Dumbledore. Would magic take away her son like her sister was "taken" from her?

In DH, Dudley finally takes a stance away from his parents' expectations which at first shocks them, but then they seem to realize that his actions fall right into their belief of "keeping up appearances", so no matter what really motivated Dudley to extending his hand to Harry, they can rationalize to themselves that they "brought him up right".

To me, it seems that Dudley was being genuine towards Harry in that scene. If his wife was a Muggle of his parents' ilk, however, it is just as easy to explain to her that Harry is family, and the occasional visit and annual Christmas Card is well within the "proper" etiquette rules - (and it also keeps an open line of communication with Harry because he wanted to). Also, by making Harry and his family "familiar" (especially with Ginny's "no nonsense" approach) if one of Dudder's own children had abilities, he could consult Harry in the best way to "deal" with them and ease his wife into the possibility of sending that child to "Harry's" school. Having cousins there would certainly help assuage any fears which his wife might have - especially if she was confident that Ginny was "normal" and was on friendly speaking terms with her.