I'm sure this scenario plays out a lot more often than any of us would prefer it to.
I know that some here might advocate that the game belongs to the first shot. I don't follow that school of thought, however. I've picked up too many gut-shot deer and elk that had made it through the night. Once you lose or are off the trail, you no longer have any claim to it unless it's you who picks it back up somewhere and finishes it off. Not law, just etiquette.
So, in this case (and if it were me), I'd not hesitate giving your uncle credit for fnishing the bear and legally tagging it (I wasn't clear on the whole "last day" thing, either. Had I not read all the way through this, I'd have said your uncle would be guilty of taking game out of season here in WY). It's a good family story though, whether there's taxidermy involved or not.
I'm one who's probably overly particular about the shots I'll take. Nonetheless, I think it's a bit premature of me to say that I'll never lose an animal. That possibility alone makes me even more cautious. But, hunting (to me) isn't at all about hanging something on the wall. It's connecting with the outdoors with an intensity I don't feel when I'm just hiking or camping - first. It's venison in the freezer - second. And if there's something to hang on the wall afterward, a decision between paying some bills and paying the taxidermist. Everyone's motivations are their own, however.