Yes, there are better ways to use browser agent id. But keeping Flash on the desktop means their HTML5 code does not need to be validated on lots of browsers. If the BBC implementation of HTML5 turned out to be buggy, the damage would be limited to platforms that couldn't run Flash anyway.
If I were in charge at BBC, I would use mobile/portable devices as a beta test for implementing HTML5. Sooner or later, they have to bring HTML5 to the desktop, but it can wait until more of the obsolescent browsers are gone. Maybe the next project is to implement adaptive style sheets to get one code base that suits all browsers on all devices. At that point, Flash can finally take its rightful place in the Recycle Bin.
When you have a huge user base and many of them are technologically illiterate, you end up doing things that are far from elegant. In a large organization, it takes longer than you would expect to get anything done.