I'm posting this in case it helps anyone else who gets frequent disconnections at the BT HomeHub. We have had multiple daily disconnections now for two and a half months and BT have variously said that: they can see a fault outside the property / there is no fault on the line at all / the line is "unstable" / they can see 27 disconnections in a day / there is no record of 27 disconnections in a day.etc.
So no progress for 9 weeks. BT have repeatedly sent Openreach engineers tasked with the wrong job three times (i.e. repeatedly check the home installation but not the cabinet / exchange) in spite of having given the home installation a clean bill of health four times now.
Finally this week we got an engineer who was superb. He immediately saw a problem with a huge number of FEC errors and the noise ratio on the line. He showed us a graph of the FEC errors on his device and we are amazed that no-one at BT / Openreach was able to see this n the previous 9 weeks. He got a "lift and shift" and, fingers crossed, the broadband has been stable since.
So, if as many people who post on here, your HomeHub repeatedly goes green / orange / flashing orange) it might be worth asking for the level of FEC errors to be checked. After all, before this engineer four previous ones failed to spot it.
I hope this gives others in the same position at least a line of enquiry with BT.