Coating regular gum would probably not work. The nicotine is embedded in the gum base itself. That's why the instructions for using nicotine gum say to chew just long enough for you to feel the tingle that nicotine creates against your oral mucosa, then tuck it between your gum and cheek for 30ish minutes or until the tingling stops. Then you give it another couple chews, just enough to expose more of the nicotine to the surface, and place it between gum and cheek again.

The nicotine is slowly absorbed through the oral mucosa. Coating the gum would give you an initial burst of nicotine, but would not have the sustained drip of nicotine that the traditional nic gum does.

I suppose one could melt down gum base, add a vaping nicotine solution, stir it together, let it set and then cut into pieces... but I would worry about the PG or VG that the nicotine is contained in doing something weird to the gum base. And you would have to be careful to stir it quite thoroughly to prevent nicotine "hot spots" in your gum.

Making a homemade lozenge or mint would be much easier, and I believe some people do, in fact, do just that. It would be similar to making hard candies, only at some point you would mix in the nicotine liquid. Since many of the flavorings used in vape juice were originally designed for use in baking and hard candies (and usually have a PG or sometimes VG base), I'm pretty certain the PG or VG of the nicotine won't screw up the texture of the candy.

I know how to make gum and it would be a hassle and a half for this purpose. I'm fine with unflavored juice composed of compounds that have been used for inhalation purposes for decades.