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I know it's commonly used as a nickname for Anna is Russia but I don't think it would work anywhere else. Just my opinion.
Nan and Nancy have been used for ages. I don't think such a nickname could be easily established today. The reason why it worked back then was because at that time there were sometimes two or three people with the same common name in a family which were usually Elizabeth, Mary and Anne so people got extremely creative and the nicknames just became familiar. But nowadays with such a great variety in names I think it would feel forced. But that's just me and I'm known for really disliking far fetched nicknames. I actually know girls named Anna nn Anya from Poland but still it doesn't work for me.
This.