If I were the guy working the mitts with the southpaw in the original video, there things I would find problematic. He is making his moves haphazardly, assuming that his trainer isn't going to bust him in the chin with the mitt. So he just leans around the 'punch' and has his chin way up in the air as he drops his hand and wings that 'uppercut.' An uppercut thrown with the lead hand is almost always going to be a counter punch. You have to slip the length of the punch and move your feet (and turn your shoulders) accordingly. The guy in the video is not doing that.
If I were teaching that move...You want to push off your back foot to move your front foot, as you turn your shoulders over that front foot. This gets you inside of and under the punch. You are now between the opponent's hands and his body. You shift the weight back and bring up the punching hand, turning the fist so that the punch lands on the two large knuckles. It is a much tighter move than is shown in the video, and your chin is behind the shoulder until you are in a place where the other guy cannot reach you.
Years ago, Tracy Spann was a murderous punching southpaw LW ranked very highly. He fought Honey Boy Paden, a slick boxer that didn't have a great record but gave everybody a full night's work. Paden had a lot of success early with a left uppercut, to the body and then to the head. And he got careless and didn't get past the Spann jab before he punched and Tracy hit him with a right hook. Almost corkscrewed his head off. Incidentally, the only difference in southpaws hitting righties and vice versa is the level of practice. Most southpaws fight more orthodox fighters than the other way around. If it works one way it will work the other if you set it up right.
If you want to see a nearly perfect way to throw that punch skip into about the 29:00 mark of this video.