I want to bring my daughter up calling me by my name, rather than Mummy (please note I am not asking if IABU about this, I am happy with this choice).
I have been very careful to find out and use the names my ex and his family wish to be known by (the two great-grandmothers are Nanny X and Great Granny, grandmother is Nanny, grandfather is Grandpa, father is Daddy).
They all know I want to be known as but insist on calling me Mummy. As I rarely have the opportunity to refer to myself by name, presumably she is going to pick Mummy up in preference.
It would be fine if they were using it as a descriptive label, in the same way as e.g. sister (your sister has a toy for you), but in any case where they would use the sister's name, I would prefer my name to be used.
I haven't been pushy about this, I stated it before she was born, and a couple of times since when relevant (i.e. they were talking about when she might learn to say the word).
I think if my ex would go along with it, they would too, but he is abusive and not likely to agree to anything unless there's something in it for him.