Thanks for the clips.
I find Mary's mastery of Danish in regards to her vocalabury and grammar to be near perfect.
She is also more articulate than Frederik. On par with QMII and Joachim in fact.
She has a very distinct accent, which I don't think she will ever lose. It's no problem for me as I can easily understand her most of the time. It happens I have to pay a little attention, when she is talking. Mostly, I guess when she is mentally going through what she want's to say or convey. When she's just chatting away, no problem.
I think many finds her accent charming.
I don't recall anyone making fun of her because of her accent. Making parodies, yes, of course, but that's another matter.
Also, I understand that Danish sounds are pretty difficult for English speakers in particular.
ADDED: I agree with FasterB, Mary is clearly thinking in Danish most of the time.
She isn't anywhere near getting the same treatment as Prince Henrik.
Our Marie in comparison is close in the heels of Mary and I believe she will eventually speak Danish with less accent than Mary. Perhaps she has a talent for languages or because it's an advantage for her to already have learned several languages.