It does sound reasonably like a Flash problem. As a quick test, can you please switch to the HTML5 Youtube player and see if that works:
https://www.youtube.com/html5 [click on "Request the HTML5 player"].
It's not the ideal solution as there are times the HTML5 player can't be used, just a quick test to check what's working and what's not.
Other than that, a detailed description of what goes wrong when you try to play Youtube videos is our best bet.
Richard