Gidday Nib and thanks for your post.
Wow that is certainly a disappointing and difficult situation.
I had a similar situation many years ago (although nowhere near as severe) when I saw my Grandfather for the first time in years. His greeting was 'Get a haircut!" and he turned his attention to others in my family. That put me right off further contact for a while. Later on when we met again I made a point of asking how he was, how his garden was going, and made a point of getting to know something about his local RSL to give us a little bit of common ground.
It worked pretty well and our relationship was not perfect, but much better from then onwards.
With your Grandmother, I think its always a shame to have a feud within any family - its much better to resolve it if possible. There is some very good advice in the other replies to your post, so I will come at it a bit differently. Yes, try sending her cards or a little present on her birthday to test the waters to so speak.
She obviously has a very big issue towards you, driven by an problem no doubt that she has with herself and how she regards herself. Working away gradually and thoughtfully should generate some progress over time.
The trick is to find out what she is into - interests, hobbies, beliefs and issues, then work on those. That will show her that you have actually made an effort to find out something about her and her life. It might be rocky for a while and her reaction my disappoint you, but be reliliant and don't give up!
How about this? Give the above a few goes then ask if you can have a brief chat with her. Open up, tell her you want to love her as your Grandmother, and ask what her issues are with you. I know this approach can work wonders as this may break down her barriers so to speak when she hears that you love her and want to have her as your Grandmother.
I fully realise these steps will take some courage and might sound a bit scary as well. Its very much over to you how you feel about it as the situation and relationship is pretty dire!
Very happy to discuss further with you should you wish.
All the very best, The Bro.