Dan,
It's hard to tell from your photos how your car is wired. The fact that the relay looks to have only 4 connectors would indicate that it is a non-cancelling arrangement . But the yellow colored wires seem to indicate it is using later series wire, but maybe actually connected somewhat like the SI connections.
See the wiring diagrams for SI and S3
http://sunbeamalpine.org/wp-content/uploads/2017/09/s1_diagram.pdf
http://sunbeamalpine.org/wp-content/uploads/2017/09/s3_diagram.pdf
Note that the SI (and SII also) use green wires (with different color tracers) on the OD circuit. Note that the OD switch gets power directly from the green circuit, which is switched by the Ignition switch and also fused. If your car is wired this way, then the relay should have 12 V at the relay whenever the OD switch is ON, regardless of the gearshift position or the Top Gear switch. In the SI and SII factory wiring the Top Gear switch switches the Ground path not the power source. You say you get no 12 V anywhere on the relay with the OD switch ON (and the Ignition key ON also I assume) I assume your other "green wire" circuits , like brake lights, blinkers, gauges, and windshield wiper are working so I am guessing that the green circuit is OK, but either your OD switch is bad or your OD is wired more like an S3 .
In the S3 diagram see the wiring of the OD. The power to the OD system comes from the "white" circuit,( switched by the Ign Switch , but not fused). The white wire for the OD relay goes to the Top Gear (or Gearbox) switch and from there via Yellow wire to the OD Relay. And then the OD switch controls the Relay On /OFF by completing the circuit to ground. Note the S3 uses the self cancelling relay and switch but the wiring arrangement could have been easily changed to use the simpler 4 pin relay and simple On/Off switch. If your car is wired this way, (that is, the 12 V power goes first to the Gearbox switch) and you find no power at any relay pin even with the gearshift in 4th gear, then I would say your problem is with the gearbox switch. And if that IS the problem it's quite likely that the problem can be solved by removing some shims that are used to adjust the "reach" of the plunger on the switch, such that in 4th gear the switch actually gets actuated.
Hope this helps.
Tom