For mule deer earliest I have seen is early January, latest I've seen a bigger heavy older mature buck in the bookcliffs unit still packing around mid May. Also seen a few years ago someone shot a buck out there on the rifle hunt end of October and both his antlers popped off when he fell.
My dad killed a little two point when I was a kid that both his came off while dragging him to the truck as well.
For elk... earliest is probably mid February, and latest would be mid May.
Seen a very large bull with only one side around February 1st, but couldn't make out if he had dropped a side, or if he just never grew one on that side, he was a long ways off looking through the spotter, I was fairly confident he hadn't just busted it off, but couldn't 100% tell if he had a pedical on that side. If you doubled his side he had, he would have probably been in the 370+ class. I scoured the area for a shed after they moved off but never found anything.
I do know a guy that killed a one horned bull that didn't Even have a pedical for another antler, so it's not out of the question that is what I seen as well.