The plastic parts and wires came clean with a little mineral spirits and elbow grease. I also cleaned up some of the sheet metal enclosure pieces.
Based on my success with the heater, I decided to tackle the heater temperature control mechanism which goes with it. Believe it or not, after a lot of WD 40, scraping, wire brushing, and sanding it does work!
The copper "wire" is actually a tube which acts as some sort of thermostat, overriding the manual controls. There apparently is a seal on that part of the device that needs replacing, and will need a little more investigation. Presumably there is supposed to be some sort of liquid inside the tube - but what, and how to get it in there again is a mystery. To my current thinking, it is not a critical component. This is what it looks like cleaned up.
(If you look closely, you can make out some of the manufacturer's information. This piece of the car is British!)