I don’t have a manual on the Gazelle, but on the Smiths gauges used on the Imp and Alpines the temperature gauge would default to the lowest temperature when the key is off. So, what I’m going to say may be invalidated if the temperature sensor is not a NTC type, and the gauge is not a wound heater type. You’ll have to step through each component and validate them.
Start with your voltage regulator first. The voltage regulator should be outputting around 10 volts on average. The original is a bimetal strip, so it won’t be easy to read with a digital meter. It needs to have a good ground as well.
Assuming that this temp sensor is a negative temperature coefficient thermistor, then you should be able to check it with an ohm meter under controlled conditions (out of the car, in a pot of water) using a thermometer as a comparison. That would validate the sensor. NTC sensors start with a high resistance at low temperature, and decrease resistance as the temperature goes up on a curve.
Then you need to validate the gauge itself. Smiths gauges I’m familar with are wound resistive heated. (The Gazelle may be different, you’ll have to let me know… but most cars of this time period used a Smiths gauge with this basic technology for instrumentation.) The NTC sensor allows more current to flow as the sensor gets warmer… this makes the heater in the gauge warm up, and the needle move toward the hot direction.
Without a variable power supply, its difficult to test the gauge itself. (You could always bench test the sensor with the gauge out of the car.)
Finally validate that they all work together. Basically find out if its wired right.
Starting at the ignition switch.
With the switch on there should be power the voltage regulator input terminal.
The output of the voltage regulator goes to the gauge on one side.
The other side of the gauge goes to the sensor.
The sensor is grounded to the engine and completes the electrical path.