OK, just read back & apologies for asking what you'd already answered, including that you have issues with iPlayer on the BT box also.
Just a thought but when you installed the replacement box, did you use the AC adapter that came with it or stick with the old one?
If the former, have you tried plugging it into another socket?
I'm still leaning towards another duff box, or one that may have been nobbled by the EE rebranding software.
So it could be issues with the ac adapter then. Seems daft not to supply one.
Long shot but try swapping the TV & ac adapter around, just in case for some odd reason it doesn't like the surge protection.