Have a look at this thread about FreeSat boxes and other devices to connect to Infinity. Report your problem to Dean the moderator through the link on that thread. BT have an issue here. It's just getting them to address it.
I'm not sure what happened but the problems seem to have disappeared. I hope that it was BT diligently investigating customer issues and sorting it out rather than a local traffic hog going on holiday for a couple of weeks.
Either way high quality iPlayer content is now playing at peak times directly on the Sony TV without any problems. Mrs Snowdude is very happy!
Thanks to all posters for your time.
(FYI, the TV's connected using a direct Ethernet connection (1GB switch) so it wasn't wireless issues.)