When I listened to Princess Élisabeth and Prince Gabriel's speeches at Queen Fabiola's funeral and the WW1 centennial, I noticed that their Dutch was quite distinctly French-accented, despite their attending Batavophone schools.
I think the primacy of French (or their mother's heavily French-accented Dutch) in their upbringing shows itself in their un-Germanic prosody. They don't stress the syllables that must be stressed in Dutch, but have more of the fluid stress of French. Additionally they don't distinguish well between long and short vowels, something which is essential in Dutch and inexistant in French.
I was also quite baffled by their use of a uvular r in Dutch, but I now see that the Brussels Flemish dialect (together with Gents) actually uses uvular r, although this cannot be said to be standard Flemish Dutch, can it?
Since so few criticize the young royals' Dutch compared to the massive critique launched upon the older generations, I'm starting to wonder, as a foreigner familiar with both Dutch and French, if Élisabeth and Gabriel actually speak some strange new sociolect: an artificial Dutch learned by a Francophone Brussels elite determined to be bilingual, i.e. the modern equivalent of Franskiljons....
NB - don't misunderstand. I don't criticize the children themselves, they just speak like they are taught and if all Belgians can understand them in their own mother tongue, then more power to them. What I find strange is that their parents were so determined to have them raised bilingually and then seem to have failed to give them a true Dutch accent, something which should be easy with regard to small children, who can pick up any language and accent in a way most adults can't. Perhaps the error is that they have learned too much Dutch from people who don't speak with a good Dutch accent (e.g. their parents) instead of truly native speakers?