The BBC website supports the HTML5 display of videos on devices such as tablets but currently still requires Flash elsewhere. Your post suggests that either you've not got Flash installed or that it's not working properly.
Take a look to see if your list of installed programs includes any reference to 'Adobe Flash Player. If so, uninstall it. (Instructions here, if needed:
http://windows.microsoft.com/en-gb/windows-10/repair-or-remove-programs#v1h=tab01 )
If Adobe Flash Player was listed, and you've uninstalled it, RESTART YOUR COMPUTER. (That's IMPORTANT!).
Either way, then go to
https://get.adobe.com/flashplayer/
Click to remove the tick under 'Optional offer', then click on 'Install Now'. Follow any instructions you see (which might include double-clicking on a downloaded file).
Restart your computer again. (Probably not strictly necessary but always advisable after installing, or uninstalling, anything - it forces Windows to update its registry).
Try accessing a BBC video.