I have been in this position exh left me early pregnancy claiming I cheated and dc was not his, pil sided with him and my pregnancy and early years with dc were just awful, it was so hard emotionally, physically and financially.It very quickly came out that ex was cheating, he moved in with ow and she became pregnant. Ex did ask to see dc and has probably 12 times (DC now 9)
At some point after a few years I think that it became obvious to pil that their ds had been lying about me and dc, their attitude towards me changed, I cant pinpoint the change but they really worked at it, it was a very gradual process. They made it clear that they respected me and that they thought I was doing a great job with dc and that they wanted to be a part of dc life, it has taken time quite a few years for me to be able put the the way in which they treated me aside, I cant forget it but I can and have moved on. They phone regularly, they send little gifts, silly things that show they are thinking and care about dc, they visit regularly and are now an important part of dc family. It has not been an easy road for any of us, but we have all made sure that dc is the most important person in all this.
Ex is not part of this, rarely sees dc does not contribute financially and we don't discuss him.
OP you sound just like my pil in the early years, a complete arse, if I was the mother you would not be getting near my child with your current attitude. You need to acknowledge that your family have behaved badly towards her, and accept that you can not dictate anything at all, it is not going to be easy . I would suggest a letter as a first contact, saying how sorry you are for the past, accepting you have behaved badly and made mistakes saying that you would love to be part of your grandchilds life, making it clear you understand that this has to be in a way she feels comfortable with and in a way that is best for the child .Ask if you can maybe have a picture of your grandchild. Maybe suggest a coffee in a neutral place, take it slowly.
You don't have any rights, you have no choices, a bit like her when she was pregnant and your son and you dumped and treated her like shit