If you can remove the item from the wood, I would boil in dishwasher liquid/powder and tap waterfor 5 minutes, then rinse well and dry thoroughly with gentle heat and/or compressed air. This will remove all the old oil and grease but not touch any colour hardening etc. If there is surface rust this will become very apparent after this treatment and you can decide how to proceed depending on the presence of any hardening colour. Obviously re-oil afterwards to preserve the finish.
I would strongly advice against using scotchbrite, the US version may be different from ours but my experience is that it strips blacking and browning a treat so I guess it will do the same for the hardening colour.
