This situation has reached the point where it really doesn't matter what started it. It needs to stop, before it escalates any further. If you can, I think you should try to reconcile with your partner and work out a way forward together. Yes, his mum was out of order and I would be furious for a while, but there are a couple of things you need to think about.
Firstly, you seem convinced that she is somehow trying to "take over" your daughter. Well done, you've handed that to her on a plate, with a baby thrown in for good measure. With you living apart, your partner will need childcare while he works (assuming he wants 50/50 care). Guess who he will go to for that? She may even be living in Granny's house during her time with her father.
Secondly, have you read the threads on here started by women who have to deal with ex-partners and child arrangements from birth? They rarely involve the words "calm", "peaceful" or "amicable". If you can find a way to stay together, then you will all benefit. Of course a conversation needs to be had about contact, and since you find her such a problem, you could say that she can only come round when your partner is there (and enforce it, even if she turns up at the door). He can also take both children to visit her without you, once the baby is weaned. I have actually cut my own parents off from my children, so I know it can sometimes be unavoidable, but they were seriously abusive to me and showing signs of starting on my children. This isn't what happened here.
I think you probably need some counselling, both together and you on your own. The strains of pregnancy, birth and raising young children can sometime throw perspective out of whack a little, and getting some support will help you. Depression in pregnancy is also not unusual, and it might be worth speaking to your midwife or GP about that. I say this, because you don't seem to be just a standard-issue twat, you seem like someone who is struggling and has got themselves into a bit of a state. Your partner may need to work on his issues around his mother, that isn't uncommon. But my main concern is that you seem to be in need of help, and I hope you find it.