@ Andy
I mainly use IE, but I downloaded Firefox a little while ago when WordPress introduced a major upgrade which was initially very glitchy in IE but functioned more smoothly in Firefox - otherwise I tend not to use Firefox. My son uses only IE. If the lastest version of Flash does not consistently provide a fix then I'll suggest that he downloads Firefox as well (it was his PC that was having the problem with iPlayer, not mine which has been OK).
There is a BBC technology blog thread on BT iPlayer Home Hub issues although a Zen/Speedtouch 585 user with the same iPlayer problem is now contributing. I understand that the BT Home Hub is also a Thompson product. I am still waiting to get into the blog but when I'm in, I'll also mention your suggestion of using Firefox, if that is OK with you but with the caveat that your experience has not been with a Speedtouch router.
http://www.bbc.co.uk/blogs/technology/2008/05/bt_iplayer_homehub_issues.html
I haven't been able to find anything specific to this problem with BT Home Hubs and Speedtouch routers on the iPlayer site Troubleshooting pages other than the requirement for the lasted version of Flash. Given that its been a known issue for sometime, I would have expected better acknowledgement of this issue by the BBC.