Congratulations horley, wonderful to be a new grandmother, but hard to know what to do to get it right, isn't it? I think how people react and behave depends on individual family systems. Some are more intertwined than others.
We tend to hold back, reading your post vampirequeen the word than sprang to mind was respectful and also trusting. I trust our family to manage their own children and I respect their right to do so. I'm a paternal and have sometimes felt pushed out although I have a close relationship with both daughters in law who include us in their thinking. Distance plays a part with one family, the other daughter in law comes from a family of rescuers. They all rush at the hint of, well, anything while we offer help and then wait for the offer to be accepted, much as you are doing horley and vampirequeen.
Don't worry, horley, you will find your grand parenting space. Your daughter knows you love them, she is valuing your advice. The paternal grandmother sounds a bit of a steam roller and also a worrier, maybe she was worried about not being needed and is overdoing being helpful. If you want to see your family more often and can do so then why not check with your daughter when would be a good time for her and just go? You don't say how old your grandchild is, I wonder if there is any potential for them to come to your home for a visit, and maybe some peace and quiet
. Please don't worry, things will settle down.