It's a mock of baby-talk. Grandmother, long winded as the word is for a name, has a good half dozen abbreviations. Somewhere along the lines, it was naturally shortened to "Granny". With the undeveloped mouth of a toddler, shaping a "G" is much more difficult than an "N", but even more difficult is shifting the mouth around into differant shapes quickly, as with shaping words- which is why most baby-talk is repetative "baba" "gaga" "googoo". So as they stumbled over a G followed by an N, it became easier to make it two Ns, Nanny. Nanny becomes quite commonplace, and younger mouths often learn that of their family member's names first- and "nana" is a very slurred variation of Nanny. Because family is so important in the live of an infant, most family names have taken simaler paths. Father- as we formally call it in English, derives from the word Pader. Which is where we get the branch of both "Pappy" and "Daddy", their babytalk variants "papa" and "dada" and their baby-talk-mock "Pop" and "Dad". Mother- as we know it, derives from Mater. But most ever variation (mom, mum, mommy, momma, mamma) adds another 'm' and is probably closer related to the word "mammel" than "mater". With mammae being a word for the milk-bearing organ of a mother's breast.

Grandmother come from Europe Nana come from slavery in the Americas

Easier for youngens to say.