I shall soon be a first-time grandmother (love all the tips - thanks!)
In my area, the traditional names are: Nanna / nanny & Gramps / grampy.
In my family when we became adults it was thought that these names were childish, and we were invited to call our parents & grandparents by their given names (this goes back at least 3 generations)
However, when my children became adults they laughed at the idea of doing this - we remain "mum & dad", my parents remained "nanny & grampy", and their great-grandad had a family nickname that we all called him.
My parents recently died and I suppose we would traditionally become "nanny & grampy". I myself would be quite happy to be called by my given name (a suggestion that has caused even more hilarity!)
So: I find "grandma" formal, "nanny / nanna" fine for small children (but I hope to be around until they are adults).
My children think it's silly to use my given name (as if I don't want to be a granny - I do!!)
I don't have any cultural background to enable me to get round it by calling myself "oma / nonna" etc.
Please pitch in, you all have such good ideas about everything else!