What I've found:
This started a few hours ago.
It's nothing to of with 10.10.1, or Safari (its been days since I updated)
flash installed didn't work.
Uninstalled flash, and so turning to HTML5
That doesn't work.
Installed Flash, and Click to Flash
It has intermediate effect.
Changing the User Agent in Developer Menu, to something else works.
Playback on iOS works.
It seems to be the Safari Desktop User Agent is effected by, what I presume is, a change by You Tube.
This happens. Supporting lots of Devices and browsers is getting to be a huge issue, especially as almost every connected TV now is a different user-agent.
There has been some slippage here at the YouTube end.