K.D.
Hi, D..
If your mother-in-law has moved in "to stay in" -- in other words, if this appears to be a permanent arrangement and not simply a visit of a few WEEKS -- then it is important to quickly have a candid conversation about "expectations" for living together. This does not need to be YOU setting "ground-rules" for her. It needs to be you, your husband and his mother talking TOGETHER about the family living situation, hence the term "conversation." Such a conversation can be done in an entirely loving way and does not have to be adversarial.
I recommend that you have a private discussion about the situation with your husband first. You need to know his observations, feelings and thoughts. He may not even REALIZE what you experience when you are alone with his mother. You need to be patient with him (men can be oblivious) and help him to understand your need to have some "alone time" with your baby. Make a mental list of reasons that you desire this private time with your baby before you get into the discussion with him, so that you are sure to address all of the issues that are important to you. One that I think is that it is necessary for you to "try out" your own mothering skills without an audience or interruption. No one needs a full time audience when they are experimenting with their own parenting behaviors (what works and what doesn't work) with their first child. Whether he has realized it or not, he, too, needs alone time with the baby to try out his fathering skills. MAYBE he gets that time alone with the baby without his mother chattering; maybe not. [Also, in the initial discussion with your husband, be sure that you state, up front, that you are not asking him to "do" anything about the situation. Men are doers and have a hard time simply listening and hearing us when they think we are asking them to "do something" prior to gaining their understanding of the issue. What you want is to have him hear your needs and you hear his views, and THEN agree on a course of action that the two of you can pursue together, AFTER he understands the situation.]
It seems like your mother-in-law is widowed or divorced, since she was available to move in to your home to provide child care. It is likely that she is simply "lonely" and her desire to chatter may be based on this pent-up loneliness. Perhaps non-stop talking is simply her nature. Either way, the cause of her non-stop talking is not the issue. She is certainly not self-aware and her behavior is inconsiderate of others. Her needs cannot trample on your needs.
I assure you, this situation will NOT get better by itself. If you and your husband don't discuss the situation with his mother, she may never become self-aware enough to realize that she is an obstacle to something important to all involved: individual relationship development time (for each of you) with the baby. Your family conversation is of paramount importance to your baby. A child reared in any situation where a parent (or grandparent) is "set-up" to develop resentment (whether over private face-time, or some other issue) is not conducive to the baby's healthy development or the family's well-being.
Although you did not comment in any detail, you also mentioned that your mother-in-law is not "able" to provide the child-care that originally was the basis of this live-in condition. Is the problem a health issue, or mental issue, or a willingness problem? In your initial private conversation, you and your husband need to discuss openly whatever factors are in play. Then, you can deal with any of these issues in the conversation with your mother-in law, as appropriate.
Just talk it out with your husband first; then get the family together for a conversation with your mother-in-law, to establish some commonly agreed upon expectations/protocols -- in love.
Blessings,
K.