How can elders find someone.. And how long would it take?
It depends. Elders are stressed-out busy nowadays and usually have bigger fish to fry. If they encounter a few obstacles in finding the where the person's records are, it is likely they will put that on the backburner.
Or if someone at the congregation is pressuring them, they will keep trying.
I have work that involves finding people that do not want to be found....it is not the hard.