It's more usual to speak of these structures as "hooves" than as nails.
The hoof includes the "nail" (outer corny plate), plus the cornified skin and other structures that protect the tips of the elephant's fingers.
As for their growth rate, I found a very exhaustive work (see link in the source) that provided the following data (table on page 63):
Mean growth of the horn wall of the foot (i.e. the "nail"):
For Asian elephant (Elephas maximus) = 7.0 mm/ 28 d
For African elephant (Loxodonta africana) =5.4 mm/ 28 d
So as you can see, it takes about a month for the hoof to grow 5 mm in an African elephant, and less time in the case of Asian elephants.

