It's "Donk"; from when they played a game of Pin the tale on the DONKey. She just gave him that nickname.
Mary doesn't have much time for George he spends more time with Sibby and Tom when he goes to the nursery.Mary just wanted an heir so she wouldn't lose her snobbish ways.
I think Mary does care for George and love him (she loves Sybbie too), but she certainly is more traditional for aristocratic families of the time when it comes to children than Tom and Edith are.
Now with Sybbie and Tom going to America, George will be spending most of his time with Marigold. I hope they'll be close; some suggested George and Marigold could be the key to Mary being nicer to Edith. Well I certainly hope Mary tries to be nice to Marigold.
I look forward to hearing Marigold call Robert Donk as well. Who knows, maybe George will too.
That's the way things were done in those days - the children were "presented" for an hour before their tea, and that was about it. It is no reflection on Mary (or Tom and Edith) as parents whatsoever.