After repeatedly spraying the bridle with diluted dishwashing soap it did get somewhat less oily. But the reins were being difficult and still had too much oil for riding with. So even if I was coming around about the color, they weren't safe to ride in yet.
I already did the reins over night and they are definitely dryer and a little browner (yay). I'll do the noseband after the rest of the bridle that's in there now sits overnight.