Let me get this straight. The Uzumaki clan are
distant relatives of the Senju clan. Tsunade is
the granddaughter of the Harashima Senju, the 1st
Hokage. Jiraya was Minato's teacher and was
the godfather of Minato's and Kushina's
son. If Tsunade was Naruto's grandmother then
she would know him or at least heard of him when
they first met and wouldn't be on bad term
with him. And Naruto only calls Tsunade granny
because she is old and he wants to piss her off.
we don't know for a fact if Tsunade had kids
or not!! she always left the village!! she could
have been pregnant by dan or Jyira!! when he
died!! we don't know!! they tell you cdertain
characters role on purpose!! others in suspense!!