At the time I donated a kidney non-directed 12 years ago only about one third of U.S. transplant hospitals would even receive such candidates. So once one was approved they would match her/him with a patient at THAT hospital. (In my case, that was rapid once my testing was done). Now there are many more hospitals that accept such offers, and often they are linked by computer bases to search for matching recipients among many hospitals.
www.kidneyregistry.org has a large data base and tries to match the non-directed donor with a recipient who has a willing but incompatible donor. You donate to that patient, whose incompatible donor then donates to someone else. Again the "someone else" chosen is someone who likewise has a willing but incompatible donor who will pass along a kidney. Thus your donation would set off a "chain" of many otherwise impossible donations. Setting up such a chain can take a little time, as each patient needs to be ready within a limited time span. Since kidneys can now be safely transported from one city to another for transplant, often you need not even have to be in the same State as your recipient.
I know that this does not answer your question of "how long" very well, but I wanted to pass along the information in case it might be helpful. It might be wise to ask the hospital doing your testing if they will seek a matching recipient only among the patients at that hospital or if they will take part in a wider search.
Often hospitals are NOT in a hurry with non-directed donor candidates, in order to give them plenty of time to think it over and perhaps back out. So you do have to patiently persevere if you want to donate.
best wishes,
Fr. Pat