I really feel for you and am in a similar situation. Just to let you know my story - and hopefully you won't feel so alone in all of this!
I too got on reasonably well with my MIL before my daughter was born; but when it got to the final trimester of my pregnancy she became a nightmare: Choosing her own names and dismissing any names I suggested; preparing her own nursery at home; assuming she could care for our daughter when I went back to work.
I could carry on writing forever about the entitled, annoying and selfish things my MIL has done since she became a grandparent; but I'll tell you some of the worst ones!
When my daughter was born we decided on no visitors at all for the first week; I was discharged from the hospital within 20 hours of giving birth - DH let MIL, SIL and BIL visit against my wishes: we had been home with baby for less than an hour when they turned up! MIL also did similar to yours; she turned up uninvited and unannounced every day for the first few weeks and was even trying to pull baby off my breast while she was feeding. DH gave her a key to our house without telling me; she was always turning up and waking baby and I up and one day she even snook into my bedroom, took baby out of the bassinet while she was asleep and tried to give her a bottle!
I finally blew when MIL requested to take baby for a walk in her pram; I consented and asked her to return in 30 minutes for baby's feed... she disappeared for 4 hours and turned off her phone! Baby was solely breastfed at this point and she had no bottle with her. It turned out that MIL had taken baby to a nearby park to meet up with DH's ex gf and her sister to show them "her new baby". I hit the roof!
After this incident DH agreed to take her keys off her and reduce contact; she agreed to visiting once a week, at an arranged time. But this didn't stop her silly games! Now, she bad-mouths me to everybody she knows and makes up or exaggerates any incidents that we've had. Another thing she has done is rekindled her friendship with my husband's ex-girlfriend - she hangs out with her once a week and always wants to take my daughter with her. For this reason I do not let her have my daughter unsupervised.
MIL was also convinced she could look after our daughter when I went back to work, but I had noticed that she couldn't physically cope with her - due to her age. 1 month before I returned to work - when baby was 10 months old - she broke down and admitted to DH that she couldn't cope with looking after baby for an afternoon on her own, let alone 5 days a week while I am at work. So we had to quickly find a nursery for baby and budget another £1000 a month to cover the costs. I am happy with this decision though, the money we pay is certainly worth it and I have less anxiety knowing my daughter is fully cared for while I am at work.
Our latest issue is regarding the covid-19 lockdown; she keeps turning up at our house to see baby. Initially this was through a window, which was tolerable; but now she wants us to take baby into the garden so she can watch her run around. The last time she visited - against government rules - we had to remind her to step away and maintain her distance several times. It also turns out that she has been flouting the lockdown rules; visiting elderly friends who should be isolating. Before visiting us she had been to see a friend around the corner, and was then going for a walk with another friend. I have banned all visits from now on until the government say it is safe to visit family in their homes and have a gathering of more than two people. Needless to say I am not in favour for this - with my DH or any of his family. I don't care though; I am putting my daughter first.
From what I have read on other forums; our situations are not unique. It seems that mothers of sons just seem to turn like this when they become grandmothers; they reveal their entitled-inner-witch and we are left to defend ourselves against gaslighters (usually our husbands) who try to convince us it is "all in our heads"... it isn't!
What I have learnt from my experience is that the best thing I can do is be firm in my choices, stand up for myself and not be afraid of who I upset - as long as I have my daughter's best interests at heart.