Yes, a reincarnated creature has the lifespan of its new body
The relevant part of the reincarnate spell description says:
Provided that the creature has been dead no longer than 10 days, the spell forms a new adult body for it and then calls the soul to enter that body.
As you can see, the spell creates a new adult body for the creature.
Regardless of lifespan, if a creature dies of "old age", it generally means their bodily functions failed in some way that resulted in their death. It is a dysfunction of the body, not of the mind (except to the extent to which the former causes the latter).
Thus, your new lifespan is determined by the new body you're reincarnated into, not by the age you were before you were reincarnated. Nothing about the spell says it maintains your "proportional age", so the age of your new body is simply that of an adult of that race (you're not a child, or an old man/woman). You then have the typical lifespan of that new race, from then on.
Rules designer Jeremy Crawford unofficially confirmed this interpretation when he answered this question in a series of tweets from November 2017:
More about reincarnate: if you're a 200-year-old elf, you're still 200 after being reincarnated. But really the question is about the body. You get a new adult body—not young or old—appropriate for the body's race, so your inner and outer ages can be mismatched.
Which age determines when you die of old age?
If your soul and your body have different ages, bodily death is tied to your body's age, not your soul's.
So that means reincarnate can be used repeatedly every time a creature dies of old to let them live indefinitely?
This fits with the spell description, which has no exception preventing it from working on creatures that die of old age, and which states that it creates a "new adult body" for it. Your soul and memory are the same, but it's a totally new body - and if the new race has a longer lifespan, the creature now has that lifespan. (Though how often do adventurers actually live long enough to die of old age?)