Though I (thought) I had removed all history, cookies and cache previously, I went back again this time and used FireFox's:

Tools > Clear Recent History > Clear Now

but this time, made sure to check ALL the options:

* Browsing & Download History
* Forms & Search History
* Cookies
* Cache
* Active Logins
* Site Preferences

Bingo, I'm back watching YouTube on Firefox again. Thanks to everyone for your helpful replies.

Scott