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This is now all very strange. I tried everything that had been suggested, other than not buying whatever needed to cast from my phone. Nothing seemed to work. So I gave up and for four days watched iplayer on my laptop.
I was about to give up entirely and get rid of the tv, but decided to have one more go. This time, it did load programmes, but when choosing a programme, what came up was different from the one I had chosen. I had, though noticed, that there was a freeview app, with iplayer on it, so tried that. This time it worked, loaded the programmes and the one I wanted was there.
I have absolutely no idea why it's suddenly started to work again, it doesn't load very quickly, not nearly as quick as it does on the laptop, which suggests to me that it's the tv rather than the internet connection. The tv is 6 years old, so maybe it can't handle any further software. There is no more software available, as the setting show it's all up to date. So - who knows. For the moment it's working, but I have no idea what, if any of the measures I tried did the trick.