When a mortal enters a daedric plane of Oblivion (e.g., Apocrypha, Coldharbour, etc.), do they stop ageing? We regularly hear of mortals who seemingly live in these realms for hundreds of years (if not thousands, due to time working differently in Oblivion), yet they do not die of old age.
Immortality is an incredibly difficult thing for mortals to achieve in the Elder Scrolls universe, with some dedicating their whole lives pursuing it, yet never achieving it. Still others are unwilling to make the tremendous sacrifice necessary for it, like attaining it through lichdom (or they simply lack the esoteric knowledge required). It therefore seems unlikely that every mortal we hear about living a long time in Oblivion achieved immortality on their own.
So I can only assume that the realms of the Daedric Princes extend a mortal's lifespan indefinitely until they are killed by some physical means (violence, disease, etc.), or until the Prince themself becomes bored of their presence and dispatches them in their own way.