Somewhere in November I received my new Windows Phone 8 device, the Lumia 920. I do not often watch youtube videos on my phone, but when I tried to play a movie, the Phone asked me if I wanted to search for an app in the store.
There is a youtube app for Windows Phone that allows you to play movies, which is basically just a wrapper around the mobile youtube site since Google is actively blocking Windows Phone access to the youtube api. Although I had that app already installed, it still asked me to look in the store. When I chose "yes" there were actually no apps found.
It turns out that the mobile youtube site is not working on Windows Phone if you set your Website preference to "desktop version" instead of "mobile version". Why this happens is unclear to me but if you have this problem you could check if you might have it set to "desktop version". Changing it to "mobile version" fixes the problem and you can play videos fine.