I honestly think I'd see this as a golden opportunity to rethink contact completely.
If this is the way she is going to be, if this is the grandmother she is shaping up to be, you should have a serious think about whether maintaining this relationship is going to have a detrimental effect on your child and your family long term.
I've said this before: I've seen several quite heartbreaking posts on here from parents who let this kind of grandparent carry on having contact, trying to maintain boundaries whilst believing that despite the grandparent's inappropriate behaviour, it was always best for the child to have some sort of relationship with them. In several cases, the outcome was that as the child grew older and was able to have more contact independently of the parents, the grandparent became more manipulative and controlling and succeeded in undermining the parents' relationship with the child, even turning the child against them completely.
This woman isn't going to be a good relationship for your child to have. The only way to sort this out would be a big, BIG reminder RIGHT NOW that her right to be in your child's life is and always will be totally dependant on her NOT posing a threat to your family harmony. She toes the line, now and always, or she is OUT.
I'd sit down with your DH and get him on board with this and then make a very clear reply.
Her behaviour in threatening court action is as unacceptable as it is unreasonable. Your family will NEVER tolerate being blackmailed in any way. Her saying this has convinced you both that she is not a grandmother worth having. Contact is therefore over. Got to court and good luck to you.
She'll do one of two things. A. She'll try and take it to court and be refused permission to even apply - as others have said, she HAS access! She just isn't being allowed to override your parenting decisions! Hopefully she'll do this as it will really call her bluff completely. Or B. She'll back down and apologise.
Calling her bluff in this way really is what you need to do. Otherwise this kind of behaviour will go on, and ramp up as your child gets older and can be manipulated directly. And yes, her daughter will calm and complain...and you and your DH can explain very clearly the situation and how for everyone's sake you cannot allow her to carry on with making destructive threats as she will ultimately make HER OWN relationship with your children a miserable one.
Once her bluff has been called, you're in the driving seat. Contact will be on your terms- if you allow her back into your family life, it will be on a strict understanding that if anything, ever crips up like this again then you will stop wanting her in your lives. Crucially, she will now believe you. So you can set your ground rules like mama and papa tiger, which personally for me would include NO overnights for a very very long time, and I'd be happy to say right to her face that it's because you don't want yor child being subject to manipulation without you there. Visits with you there only, visits when convenient for YOUR family, any weight throwing around and it will be goodbye.
If you are ever to pull this back into a good functioning relationship, you simply can't let her make that sort of threat and not slap it down. Your child is a baby, this is only the start. Nip the bud now.