I knew a
Nan once whose birth name was
Nancy. She grew up in Frome, must be in her 70s by now, I suppose, and much preferred
Nan. Cheerful, efficient, pleasant; green-eyed blonde. A good advertisement for her name. I don't mind it at all, though I prefer
Nancy; the only possible problem in the
UK might be the grandmother connection, though I had a gran instead of a nan so it didn't bother me. Oh ... also baby milk, I think!
I agree that Nan-for-Ann is too much of a stretch, though I do know an
Anna (in her early 50s, named after Mrs Karenina!) whose mother calls her
Nan occasionally.
Non just gives a negative impression, however hard I try.
Nonna might just work: apart from the "nun" meaning, it's still used in Afrikaans though not very often at all nowadays. It used to be a term of address used by the servants to the daughter of the house, like "
Missy" or "
Young Madam", and the more relaxed form was the diminutive, Nonnie. The o is like RP aw, which might baffle our American readers but should work in a Welsh environment, I imagine.