Apologies if I'm wrong, but it sounds like you're not familiar with the User-agent: header.
All browsers identify themselves to Web servers with a user-agent string that identifies the browser, version, and platform (OS). Unfortunately Web sites (such as bbc.co.uk) often just look at that string, rather than checking whether the browser actually has the capabilities the site needs. And they may not even be consistent from one page to the next at the same site!
Thus, it's sometimes necessary to override the user-agent string (sometimes called "spoofing") to report a different version, or even a different browser entirely, in order to get particular web sites to work. And it often takes quite a bit of trial and error to find a user agent that will make a particular Web site work. You can find more info at this thread:
Of course, the user agent isn't the only possible reason that a video might not play. For example, it could be using a new codec (such as av1) that Adobe Primetime doesn't understand. If that's the case, overriding the user agent won't help - but it's usually a good place to start.