Quick test for gauge function - earth (ground) the sensor wire with the ignition on, engine not running. have someone watch the gauge. It should climb RAPIDLY. If it does the gauge and wiring are good. Change sensor. If not gauge or wiring faulty. Don't leave the wire grounded for more than about 5 seconds - just long enough to see the needle get to about 3/4 scale. There is an overheat light in the red end of the dial face - similar to the low fuel light in the fuel dial face.
Good - Fast - Cheap: Pick any two.
E23 735i; E30 318i; E32 735iL (X 2 - 1 White and 1 Light Blue); E34 535i; E38 735iL; R50 Cooper; R55 Cooper Clubman.
E21 318i; E32 735iL; E34 535i; E38 730iL; E53 4.4i