
Hi Lorripop,
Generally, if nails are broken down to the quick (if it is?) it's best to get all the bits of the broken nail around the outside trimmed away, otherwise they can be uncomfortable and cause more licking! You've done the right thing by wrapping it initially to stop the licking, but keeping it covered all of the time can lead to it getting hot, damp and infected - especially if he has already chewed it.
If he is lame even some of the time and still trying to get to it then I would pop to the vet and get it checked if it's obviously still bothering him. They should be able to trim away any loose nail causing irritation, give it a good clean and dress it really well if necessary. As there is often quite a bit of dirt from the ground and bacteria from saliva where they have licked then these things often become infected and inflamed. In fact if you're not familiar with nail maintenance yourself I'd get to the vet to check all's well.
For now, I'd be inclined to boil up some salt water and then when it has cooled bathe the paw in it, then dry and cover for tonight (I wouldn't use plastic unless going outdoors as this will keep the wound warm& wet which is perfect for bacteria)
HTH?
Best wishes for him.

Ooops! sorry Its late and I skim read the last part of your post, appologies! I see you're going to the Vets :-)