Update
I got all of the BBC VIDEO tags working that I am aware of (all of them YouTube video types). See 4 posts above for examples of all known linking styles working.
If you come across a VIDEO tag that doesn't seem to work, or that links to a video other than one hosted on YouTube, please let me know and I will get it working.
Now that I have a hang of making custom BBC code parsers, I will get the MENTION tag working next.
Edit: Apparently YouTube's embedding technique requires JavaScript to work properly. I wasn't planning on using more than a tiny bit of JS on this site, and certainly wasn't planning on making users run JS from other sources, but such is life. I apologize for that, and those who want to avoid it, you're welcome to install any of the many good JS-blocking plugins for your browser of choice.
If people are really concerned, I could add a per-user option (modifiable in the user profile section) for people to turn off video embedding, and link to the videos instead.