I think Nanny would be the Tom Bombadil sort. It just wouldn't have a hold on her mind, she'd stick it inside one of the hideous souvenir mugs her family brings and the forget which one it was in. She's having a perfectly lovely time as it is, she doesn't need to be a "Queen" or anything. She has more fun just being Nanny.

And then Greebo, chasing some critter, would knock over the mug and swallow the ring. Hijinks would definitely ensue.