How to cut away ceramic from a sink that doesn't fit
Since you are intending to remove the "stem" of the faucets I would concentrate on that instead of the sink itself. Seems to make sense to concentrate on the goal instead of getting sidetracked with these other details. (I have to assume that removal of the faucet is all about replacing it with something new). 

To that end I would still recommend a Dremel tool with a cut off tool and a decent pair of eye protection goggles. Use the Dremel to cut completely through the retaining nut on two or three sides if possible. See cut areas as suggested below on the flats of the nut so there is less to grind through:



Once cut through you can use a flat blade screw driver to widen the cuts and pull portions of the retainer nut free from the stem threads. Since we are assuming replacement here the fact that the Dremel cut off tool will grind into the stem threads is of little concern.

When you go looking for a faucet assembly look carefully for one that has a long threaded part that will extend down below the lip of that butchered sink. This will allow you to slip some tubing (even a short piece of PVC pipe would work) over the threaded part offset where the new retaining nut and washer will tighten to. From the looks of it the tubing should be able to clear the butcher slot cut in the sink lip. If the PVC type tubing has too thick of wall a piece of brass drain pipe would work too. Just make sure the tubing is cut nice and square on each end.