Having bought a brand new Sky HD Plus box from an Ebay retailer and found the problem still persisted, I eventually located the remedy - the cable from the Sky dish which fed through the wall via a junction box to the Sky box had an intermittent fault which probably meant that rainwater was getting in and causing the cable to short-circuit. So I paid an aerial installation company to give me new cables, a new hole in the wall and a connection to the Sky box without any intermediate junction box. Perfect now. I still can't see why I would benefit from Sky Q.
You say recordings were only ever intended for time-shifting. Yet there are detailed instructions for how to archive your Sky HD recordings via a Scart lead, if you still have one, to a recording device with a Scart connection, if you have one. I don't care that the quality of the recording is not razor-sharp.
