A Horcrux is a part of a person's soul (in this case, Voldemort's soul) and if Nagini died, her soul would be gone, but Voldemort's soul would still be inside her.
Perhaps, as she is a Horcrux, she is immortal
Nagini was immortal of age. She contained a part of Voldemort's soul, so as long as Voldemort would be alive, she would live too, unless she would have been destroyed with Fiendfyre or basilisk venom or having her head sliced off. That's actually the thing that had happened. Neville had cut off Nagini's head. Although Nagini must have some special magical traits aside from extended life.
One must wonder if she had had her head cut off with something other than the basilisk venom-impreganated Sword of Gryffindor if a different outcome would have resulted.