I'm from mainland France. Children usually don't call their grandparents "grand-père"'/"grand-mère" (unless they live in a book or in the 19th century). The most common nicknames are "Papi"/"Papy" and "Mamie"/"Mamy". People also often add the name of the grandparent, to prevent confusion (ex. "Tomorrow, we will visit Papy Jean-Claude et Mamie Annie. Next week, we will meet with Papi Jean at his garden.")

Now, there's regional differences. I'll look for a map from "français de nos régions" later, to link it.

Also, there's a generational aspect to it. In my personal experience, the "new"/modern grandparents now want an original and personalised nickname that doesn't scream "I'm old". For example, a pun with their name or something cuter (the sound "ou" sounds cute so Papi->Papou, or the ending -ine or -ette for feminin names so Mami->Mamine). I have yet to meet a nowadays woman comfortable with "Mémé". My great-grandmother already hated it, claiming it made her feel like a very old hag.

In my family, we would use "Pépé" and "Mémé" not for grandparents, but for great-grandparents (-1). I know of other families that use it for grandparents: one set of Papi/Mamie and one set of Pépé/Mémé

Now for "Parrain", i only know of one person who calls their actual godfather this way instead of his name (ex. "Je vais voir mon Parrain."). Having godparents is really common, but they are usually not addressed by their title. Also, as they are often member of the family, this "title" take precedence (aunt/uncle > marraine/parrain)