I know just a little bit from my uncle. He's waiting for a new heart, and I can tell you about his process.
http://www.unos.org/ UNOS is a big non-profit group to help network doctors, patients and organ donors to help organize transplants all over the country.
It's taking my uncle about a year to get on the waiting list. He had a surprise quadruple bypass in Feb. 08. They had to kind of exhaust the other heart options first. Open heart surgery didn't help him, his pacemaker is giving him some borrowed time, etc. He had to wait six months after his surgery before he could meet with the transplant coordinators--that was in August 08. And he's scheduled for a psychiatric evaluation in 5 days. If that goes well, they'll put him on the transplant list on Jan 13 09. They gave him about a 4-5 month estimated waiting time for a new heart. He's very sick and was not supposed to even make it through surgery last year, so I suspect he's somewhat of a priority.
http://www.optn.org/policiesAndBylaws/policies.asp Priority depends on the organ, depends on who's a match for the donor, and depends on the patients in the waiting pool after they determine who all on the waiting list would be a good match for the donor. That link breaks up some pdfs into individual organ information.
Hopefully some of this is helpful...
JD
