I love the history of it and do like the sound, but it kinda reminds me of an even shortened version of Nanna for Grandma. But it wouldn't really be a big deal if you were calling her that at home (and even if the pet name caught on in school, etc. it'd be fine. It'd be pretty obvious that she wasn't a granny).
Btw, a literary connection (and I'm sure there are more)...isn't one of the girls in L.M. Alcott's "Little Men" named Nan, short for Anne as well?