If a creature is true polymorphed permanently, do they age in their permanent polymorphed form and possibly grow old much older than their original form could have? Or after a natural lifespan do they die immediately or reverted to the age they were before they got polymorphed?
The rules do not say, though it seems sensible that the original form would not age and the polymorph form would.
Unfortunately, I don't think we're going to get a clear answer here. You have asked this question because the spell description does not provide any meaningful insight, nor do the rules talk about this anywhere else.
However, if we may engage in the danger of applying logic to the rules of D&D, I think we can get at a satisfying ruling. When under the effect of true polymorph, we have:
The target’s game statistics, including mental ability scores, are replaced by the statistics of the new form.
Everything except for alignment and personality are replaced by the statistics of the new form. That is, everything you were is gone, you are now the new creature (except you keep your personality and alignment). From here, it seems reasonable enough to say that you age according to the new creature's aging, since you are the new creature. Further, since aging is generally the result of physical processes, it seems reasonable to say that the original form does not age because it does not exist while you are polymorphed. All the processes, the exposure, that leads to the effects of aging is not happening to your original form, while it is happening to your polymorph form.
In the end, it is the DM's call, but I think this is the most reasonable ruling on the situation.
Now, what happens if you die of old age while polymorphed? I don't know. It's up to the DM. Assuming the ruling I outlined above, dying of old age would drop you to zero hit points, which would then cause you to revert to your original form, which has not aged since you were under the effect of true polymorph. But again, this is well within the realm of DM ruling, not written rules.