Possibly coined as a variant of nanny or from baby-talk. Many Italian-Americans use "nonna," "nonni," or "nonnie" for their grandmothers.
Meanings
(diminutive) Grandmother.
Nunny is what we call our grandmothers. [...] In contrast, Nunny for my niece is my mother. [...] Her Nunny smiles broadly in her squinted, green satin evening suit… 2006, Carol Birch, Whole Story Handbook, page 37:
The greatest influence on my childhood faith was my maternal grandmother Elizabeth, my "Nunny." She and my grandfather lived across the street from us. 1994, Muto, Susan Annette & Van Kaam, Adrian L., Divine guidance : seeking to find and follow the will of God, page 18: