I don't think so since Kushina told him that the Uzumaki's were wiped out during the 3rd great ninja war. Even tho this was after he met Nagato, he probably never connected the two.
As far as Naruto is concern , there is not Uzumakis left he doesn't even know Karin is one , i doubt he knows that Nagato was one either , hell Nagato probably didn't know .
Kishi has to tell Naruto at least that , I mean you have someone from your clan right there at the damn battlefield , that is pissing me off .
I don't think he knows most likely all the other uzumaki's don't carry there last name since most of the survivors went into hideing narut's mother was selected to be the new host of the nine tail so most likely it was before the village was destroy and she came with her family since naruto's father learned sealing jutsu from her mother. So mostly the leaf is the only place with history of the uzumaki since they retain the uzumaki clan symbol.