Just remembered that I meant to give a big public thank you to this company:
http://www.aerialservices.co.uk/
even though it's a bit embarrassing. We'd been putting up with a dodgy signal for months, which generally came back with some jiggling around with the connection, but eventually we called in some professionals. It did not take the guys from this company long to identify a kink in some HDMI cable, and a likely bit of wire snapped inside it. It so happened we had some replacement HDMI - someone before had presumably suggested we replace this, but we'd not got round to doing so. Anyway, it worked immediately, and they didn't even bother to charge us.
I don't know why, but they clearly know what they are doing, and are anything but rip-off merchants.