The Doom of Men and the Ringwraiths

[The Saucepan Man]

Maybe Sauron could "delay" their aging and deaths by binding their spirits to the wraith world.

There was no question of them "aging" in the normal sense, since their physical bodies had long since passed away. In that sense, they were already dead (or perhaps undead is a better term). It was only their spirits that remained. Being creatures of the spirit world, they were invisible when not cloaked in their black robes (although they still gave of an aura of fear when invisible, moreso in fact than when robed).

But it is correct that Sauron was not able to give them immortality. He was able to bind their spirits to the Rings that they wore so that their spirits remained in Arda, but only until the Ruling Ring was destroyed and the power of the Nine Rings was broken.