Reincarnate causes the new body to be an adult body for that race
...the spell forms a new adult body for it and then calls the soul to enter that body
So, whatever adult means for that race will be what body you get.
Any additional refinement or details on top of that (such as, what precise age the body is) must come from a DM.
Does the human immediately die of old age? - No
It is a body's age that determines when something dies, not the age of its soul. In this case there is a difference between the two.
Jeremy Crawford, lead rules designer for D&D, agrees:
If your soul and your body have different ages, bodily death is tied to your body's age, not your soul's.
Have they just reached maturity? Could they be a child? Could a dead baby get reincarnated into an adult body?
Regardless of the age of the dead creature, the spell says:
the spell forms a new adult body for it
So, it seems that RAW would indicate that a dead baby or child would be reincarnated into an adult body just the same as an elderly creature is.
The soul is always the same age/maturity as it was when it died.
DM can always overrule and have the body be younger if it makes sense
There may be cases where a DM would consider altering the effect slightly to allow for a new baby/child/infant body if that makes sense for their game. And that is perfectly within the rights and power of the DM.
Honestly, the idea of a baby reincarnated in an adult body is pretty terrifying/horrifying in general. At my table, I would houserule that babies come back as babies in an instant unless there was some compelling story/fun-based reason not to.