It turns out that the reason that iPlayer TV downloads aren’t working is not that the BBC have changed anything. It’s just that their system’s broken down again for the second time in a couple of weeks. A page on the iPlayer Help system currently announces:

There is currently a problem with the availability of content on BBC iPlayer on iPhone, mobile WiFi and 3G.

My downloader emulates an iPhone, so if you can’t download something at the moment, that’ll be why.