I have been where you are - my IL's wanted my baby to call them mama and papa, which I couldn't bear the thought of and refused to allow. If she is calling herself mummy by accident, then let it slide, but if she is doing it on purpose, then you will have to reinforce that she is Grandma/Nanny.
Relationships get really fraught when you have your first baby and even when you have happily got along with your ILs for years, a first grandchild does change things - you feel very territorial and they often want to take over (because they adore this child, which is their flesh and blood) and you clash.
It's often easier with your own parents, because you feel the bond to your own mum and dad, whereas, your ILs are not family to you in the same way (although they are to your dh, so you have to remember that).
I handled things badly when I had my first - I was nowhere near as tactful as I should have been in telling them to give me some space, but by the same token, my ILs tried to take over and impose their wishes on me and it drove me batshit crazy. We all could have done better,I suppose,but none of us were prepared for how we would feel.
Personally, I wouldn't allow my mil to look after my baby, given what you have said regarding her physical ability and the state of her house - your child's wellbeing is the only thing to be considered when deciding child care - not her wishes ( your baby is not there for her entertainment - her wishes don't come above the baby's safety). I didn't let my ILs look after my baby (they had a very jealous alsatian and no concept of child safety).
But I do wish I'd been a bit more tactful in how I refused their desire to babysit. At the time I was very emotional and felt they were trying to control me, but I think we'd have had a better relationship if I'd been a bit less blunt and phrased it more kindly.