keep abrasives OFF the ways - the parts where metal slides on metal.
Scotchbrite
is an abrasive. Packaged in a different form. It comes in different grades, from aggressive to fine.
Using very fine grade to polish what has already been cleaned, a bit of polish, is likely not going to make anything
go wrong with an old lathe.
For the painted areas, degreasers, as mentioned which are alkaline based are fine. You can find them in most of
the low price stores, Family Dollar, Dollar Tree and the like here in the USA. That will take most of the grease off.
Keep in mind, IF you take the grease and oil OFF, you will need to PUT IT BACK where it belongs.
For tougher stuff I've used spray oven cleaner, which is Sodium Hydroxide, aka Lye.
SOME of the things sold, like Whitewall cleaner spray also contains Sodium Hydroxide.
For removing paint, the older paint, like on this lathe, gives up easy to the newer "green"/non-solvent cleaners.
Back in the day it was Methylene Chloride, which is NASTY stuff...
I'd say you're best to start with a simple cleaner/degreaser spray - Fantastic or 409 are mild but effective, and
see if you can get the gunk off. Graduate to tougher stuff.
For some things an acid will be good. That ranges from Vinegar + Salt + water bath to some bathroom cleaner products
that have acid in them to Muriatic and Sulfuric and Hydrochloric acids in stronger concentrations. The commercial rust removers
& converters are more pricey and may be best in some cases.
You can take the thing out, or apart and hose it down, you just have to get the parts that are important dry
and re-oiled before they rust...
Cosmetics are somewhat secondary to actually running properly, and making sure it is worth doing a
restoration and/or painting...
Department of free advice.
Post a picture and it's easier to see what needs to be done?