I think BB is your first baby. Don't worry about you BB. Not all the mother-in-law willing to look after the Baby.
You have to think she is grandmother of your baby. She should be love her/him (even you and your husband) very much. Otherwise she will not promise to look after your baby.
Sometime the method of take care the baby may be different, but you can discuss with her.
You just think about more good things. I think you should be alright.