A Openreach engineer turned up, spoke English, didn't read from a script, listened and understood. Hoorah!
The tell tale is that my broadband disconnects resulted in a static red light. Apparently a flashing red lights means the signal down the line is broken, a static light means a broadband signal is received but the router can't work out how to connect. The frequent wifi disconnects also inferred a router problem.
He ran a few checks on the line, replaced our wall box (he had spares in the van), replaced the router, checked the incoming connection box on the outside of the house. He was here for an hour.
Hopefully the new box will sort things out. Incidentally the box BT Home Hub 5 (Type B), we had a type A before. The web interface looks slightly different.
So it seems the key is to complain about groundhog day to the distant voice in India, and they'll book an engineer visit.