The Collaborative International Dictionary of English v.0.48:

Nanny \Nan"ny\, Nannie \Nan"nie\(n[a^]n"n[y^]), n.; pl.
   nannies.
   1. a caretaker for a child; a child's nurse; a nursemaid.
      [PJC]

   2. Grandmother; -- a child's word, used especially as a form
      of address. See also nana.
      [PJC]