I can only answer for cornea and eye donation.
The FDA does all oversight (no pun intended) of organ disposition.
The main organs—heart, liver, pancreas, lungs, etc.—are transplanted within hours. They are just not viable if they are not being perfused constantly.
The other tissues—like bone, skin, tendons, etc.—do not need to be transplanted immediately. But I am not sure on the regulations of when they need to be transplanted.
With the eyes, there are four tissues that can be recovered.
We recover whole eyes for research and education purposes. These usually go much faster, but we can hold them up to a year.
Conjunctiva can also be recovered; conjunctiva is a clear covering over most of the eye (it is what gets irritated when you have pink eye). I have been working as a recovery tech for five years, and our office has not had a request for "conj" in all that time. I believe it is mostly used for research, but I could be wrong.
Sclera is the white area of your eye. It is fairly thick and flexible. If you have ever touched a reptile egg, that is what it reminds me of. We recover sclera for transplant. They use it for several things, but mainly to patch punctures. Similar to if you pop the inner tube of your bike and repair it. Sclera can also be used to repair ear drums. We can hold on to this for up to a year.
The main thing we recover is corneas. In the U.S., we must transplant these within seven days of recovery. (Recovery is usually within hours of death, but we can push it up to 20 hours after if needed.) Sometimes we have more corneas than we need, and then they are shipped overseas and transplanted up to 14 days after recovery. There is no real different outcome with the later transplant time, but the FDA in the U.S. made the rules. (You can sign up to be an organ, tissue, and eye donor here.)
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