My mum and dad are Grandma and Granddad, my in-laws are Nanny and Granddad, when we have to distinguish between the granddads we say their surnames or "Granddad with Nanny!"
They chose what to be called, I think, and it's not so much that you tell your kids what to call them but they will obviously call them whatever you say from a young age, unless they make up their own name. My nephew calls his 'other' granddad "Grandad Whiskers" because he has a beard!
My mum chose Grandma because that was what I called her mum, also to them Nanny sounds like a person who you pay to look after your child, not a grandparent, and also my Granny (my dad's mum) was still alive when my kids were born, so my mum wouldn't have called herself that.
When I was young I had Grandma and Granddad and Granny and Granddad.
MIL (actually DH's stepmother) already had several grandchildren by the time we had our kids so she was already known as Nanny, don't know if her choice or her kids or grandkids.
DH's mum died before the kids were born but we refer to her as Nanny Ann.
I like Oma and Opa but think it would be a bit odd to use unless you come from a country that use it.