Never tried it on the Healey, but unless anyone comes back with more specific information, try what I did on a home built Lotus 7 replica:-
With the battery disconnected for safety, undo the sender from the tank and take it out. With the power connection connected, [you may have had to undo this ti take out the sender]temporarily wire the sender body metalwork to a good earth on the car body work. Reconnect battery. You should get some sort of reading on the fuel gauge. Move the sender arm over its full travel and see if you get a full travel on the gauge needle. If you do, then there is nothing wrong electrically and you know that the sender variable resistor is a proper a match for the gauge. In which case your problem may be that the float arm on the sender is not the right length or maybe needs bending. That would be trial and error.
If you do not get the full scale on the gauge then the sender resistor winding is not giving the correct resistance for the gauge. Talking generally, and not knowing Healeys that well yet, this was sometimes a problem when the manufacturer changed gauge model and/or supplier. Let me know how you get on - if it turns out that you have an electrical mis-match I could suggest some tests to do to find out what resistance the gauge is looking for at zero and full scale.
good luck
Mike 56 BN-2