I'm curious about this and not sure how to respond since I've never worked on a TR3.
This gauge system should be the same as other Smiths/Jaeger stuff used prior to the addition of the voltage stabilizer. On the cars I've worked with that don't have the stabilizer you do need to insure the gauge case is grounded or you'll have problems. Sometimes the ground connection was simply through the metal dash. Certainly try Dave's suggestion to remove the ground wire at the gauge, but if it doesn't fix the problem... clean the connection points and put it back. I don't believe the gauge should "float" relative to ground.
Another curious thing to me is that you say that disconnecting either wire (from the sending unit?) makes the gauge read empty. My experience is just the opposite. If the gauge case is properly grounded I've seen the fuel gauge read "full" when either wire is pulled off the sender. This again makes me question the ground connection to the gauge case.
I don't know TR3s so perhaps something is different... but the Lucas wiring colors sound the same as does Dave's description of the power connection. Whatever you find, please post your findings.
EDIT:
Sorry... I forgot that I put together a PDF to help a Mini owner friend of mine. The PDF was written to show how the pre-stabilizer fuel gauge responds and works. It may not help, but take a look at this PDF.
https://home.mindspring.com/~purlawson/files/OldFuelGauge.pdf