Another possibility would be to get a spanner big enough to fit over the shaft but smaller than the mandrel, hold the spanner with a thick cloth ot glove and belt it, close to the shaft as you can. Even to pad the space between the spanner and the headstock with some hardwood or metal and knock the spanner handle towards the headstock to try to drive the mandrel out?
When, not if, you get this basket (or a word similar to that) out the first job is to clean ALL of the oil out. Thinners of some kind, but get it as clean as the proverbial babys bum. That way you won't get so bad problems. The oil is doing the job of a very good gripper at the moment it would seem..
Good luck lass!