If you do nothing, the quantity of black sludge will just increase until it slows and finally blocks the flow of hot-water through the pipe.
You stop more sludge forming by putting enough inbihitor in the sealed radiator system water to stop the corrosion occurring - your plumber isn't very good if you weren't even told that.
If the radiators left are showing no signs of leaking (pin-hole sized lumps of rust start appearing on the outside just before) you don't really need to change all the rads now.
Leaving the black sludge in there won't cause the existing corrosion to get any worse (provided you put inhibitor in), but you really need to get rid of the existing sludge using a dissolving chemicals, then flush with clean water several times, then insert inhibitor.
Quite a simple procedure though time-consuming. It doesn't need a costly plumber to do it - a handyperson who understands the issue would be perfectly capable.