I think these kids should be called nannykin -- these kids are her nannykin or her nannykins.
May I suggest: sitees?
If the babystters "sit" for babies, the babies must be the "sitees".
Patricia
Don't forget the classic name for children under supervision: charges.
Of course that may not capture the personal nature of the relationship. Maybe you could pun on that by calling them "depth charges?" or "close charges?" or "kid charges?" or "tender charges?"
Good luck with your equivalents. (Watsons is just too obscure!)
pjf
A nanny's charges are, of course, his/her "nanettes"!
How about "sit-children" as in step-children. E.g., this is Diane, my sit-child.
Also, the specific caller said she really loved her sit-children (see, the phrase is being used already!) and that she wanted a more definite term. For this special cases, how about "love-sit," as in: this is Diane, my love-sit-child.
All the best from an avid listener,
--dave