I've resolved my issue. After starting this thread I realised that I had my dad's laptop here and he also runs Opera 12.16. I tried playing the embedded video on his computer and it worked, as did the other embedded videos that I've been having a problem with. So, I backed up my bookmarks, speed dial settings etc. and did a completely fresh install of Opera. Even though I am back to where I was before, with the same settings and same extensions etc., the embedded videos now play on my PC too. I have no idea what caused the issue and nothing I tried, outside of a full uninstall/reinstall would fix it.
Opera for Windows