Tungsten carbide cutting/drilling tools aren't pure WC, they consist of tungsten carbide particles in a metal matrix, typically cobalt. Looking up the chemical properties of cobalt, I found that its salt can form ammine complexes, which are water-soluble. So it might be possible to etch it using a mixture of ammonia (which will form the ammine complex) with a strong oxidizing agent, like hydrogen peroxide. The solution will bubble like crazy, and the bubbles will impede the progress of the etch itself -- but it might be possible to etch the drill enough to loosen it up so you could pick it out.
Iron doesn't form ammine complexes so it shouldn't be attacked. But the peroxide could cause rust to form. Putting a clay dam around the hole would confine the etchant some, and also would be a reservoir so you wouldn't have to put fresh etchant in the hole so often.
We used ammonia-based etchants in our FA lab to etch alloys like titanium-tungsten, but in that case the ammonia was just acting as a base to react with the tungstic acid, the reaction product between tungsten and peroxide. Sodium hydroxide would have done the same thing, but sodium is a HUGE no-no in a semiconductor fab situation. Even though our lab was separated from the fab, everyone was just plain paranoid about using the stuff.
Since copper also forms ammine complexes the ammonia-peroxide etch works on copper as well. But don't use it around gun bluing, it does a number on that, too. So in your case it depends on what the item is.
