If you are curious about the science, it works because cyanoacrylate cures when exposed to moisture. So dunking in water makes it almost immediate. Same reason you can glue your fingers together quickly. The moisture in your skin cures the glue fast. Gluing something really dry takes longer...
Yes, any super glue (not just the name brand) will cure almost instantly underwater. It is just easier to get it to stay where you want it if it is thicker to begin with.
Super glue will cure instantly if you put it under water. So maybe the trick is to glob a ton on his nail, then dunk his foot in a bucket of water. Will work especially good if you can get the thicker "Gel" kind of super glue. Anything that contains cyanoacrylate will do the trick.