Since requesting help or advice, I have noticed that there seems to be a tracking or downloading bar at the top of the YouTube page. When 'tracking', it either continues with the completion of the bar, in which case the video requested plays. I this 'tracking' procedure is stopped part way, usually at the middle section, then the 'install latest version of FlashPlayer' appears.
Does this make any sense?.
Using XP and Firefox. I haven't yet tried it with other browsers, to see if there is a solution there?.