I have been advised by several luthiers and repair techs to stay away from steel wool (I was asking about cleaning fretboards and frets). No matter how gentle or careful you are, you will leave behind micro particles of steel when you use it. These can muck up your pickup and electronics on your uke. Even if you don't have electronics on your uke, it can transfer to your ukecase, your clothing and the place you did the work. Which can transfer when you handle another uke with a pickup. Controls and battery contacts are very vulnerable.
They suggested using very fine plastic abrasive cloth or paper. Like those sold by StewMac. No idea if the super fine levels of grit are available with the plastic/non-ferrous abrasives.
Be very, very careful with water around the unfinished wood on your uke. Inside the body is usually unfinished raw wood. So if you're working around the sound hole be careful. Also, the bridge and the fretboard are usually not finished. Though they are sometimes dyed to make the black more uniform. That dye is not waterproof and can run. Unfinished wood can soak up water very quickly and swell.