I found this video while exploring the net for Trek material I hadn't seen before, and I sure hadn't seen this one or anything like it! It looks to have been made after Star Trek VI, when TNG was on the air, (or at least while the first TNG movie was being made) since Scotty is in the Enterprise D Engineering room. Enjoy!
Loved that the transporter accident put Kirk's head on the female cadet's body... which he seems to enjoy a little too much! And hearing Shatner use British slang; "I don't need a brolly, you wally!" Which in English would be "I don't need an umbrella, you idiot!" Funny stuff.
Update: thanks to some comments, we have narrowed it down to 1989-1990. I would have thought it was a bit later, due to the digital effects in the head-replacement bit, but I guess they could do a good job back then as well!