So, a disease caused by Contagion or Greater Contagion takes affect immediately on a failed save. Can these diseases then be spread by the original victim and does the onset time change?
I would say that from the time they fail their save against it, it acts like a normal disease, since that it says that you use the diseases' listed frequency and DC for further effects.
So, yes. They can spread them, and the onset time is changed, as the spell is no longer in effect.