That connection has been like that since the heating was installed nearly 30 years ago, the issue now is there is no power in the cables.
Badly made connections can eventually go open circuit, as the surfaces where the wire is just barely touching oxidise over time. If you were checking for presence of a feed by touching your meter probe on the screw, loss of continuity would cause a zero reading, as that is probably the incoming feed from the programmer. (I expect the stat output is the yellow core oversleeved in red)
We seem to have three different symptoms with no certainty of how they relate to each other.
- Room continues heating when it shouldn't
- Pump runs when it shouldn't
- No feed to room stat
Definitely need to see inside the wiring centre and, if it's not obvious from that, the configuration of motorised valves (Y-plan, S-plan etc.)