You first have to ascertain which system you have fitted in your Land Rover. There were two totally different systems used by Land Rovers and they are not compatible and a previous owner could well have mixed & (miss)matched parts.
Series 2s and early 2As (before the black dash and negative earth came in April 1967 with chassis suffix D) used the moving coil principle.
Moving Coil GaugeThis type of gauge uses magnetism to move the needle. Inside the gauge is a coil which is connected to the supply terminals. When a voltage is applied to the terminals current flows through the coil creating a magnetic field. This field causes the needle to be moved by a varying amount. As the field increases the needle will move toward the maximum value indicated by the gauge. The field is linked directly to the voltage applied and the current flowing through the terminals. This value will depend upon the state of the sensor that is being used. Effectively the gauge is working as a simple voltmeter but “calibrated” as Full or Empty. This fuel gauge should work with either positive or negative earthed vehicles with no changes being required.
From 1967 on Land Rovers used the bi-metallic strip gauge.
Bi-Metallic Strip GaugeThe bi-metallic gauge operates using heat. Inside the gauge is a small piece of bimetallic material that is connected to the needle. Wrapped around this strip is some thin wire which gets warm when electrical current flows through it. This material is created using two different metals bonded together. These metals expand at different rates when they are heated. This causes the material to bend as one metal expands to a greater length than the other.
This gauge needs a constant voltage supply to maintain “accuracy”, this is supplied by a unit call the voltage stabiliser which puts out a constant 10 volts, no matter the state of the battery/charging system.

Looks something like this, a metallic canister about 1” long, usually mounted on the back of the speedo and also “works” the temperature gauge.

You can see a neat animation of how this system works
HERE The Tank SendersRegardless of the gauge used the type of gauge the fuel tank senders are basically the same. You have a simple float arm that operates a wiper on a resistance wire. This causes the voltage to vary (or the current in the bi-metallic case) which moves the needle on the gauge.
As mentioned before, you can’t mix & match gauges and senders…

This is the S2 early 2A gauge/sender - moving coil type.

Here is the later 2A setup - bi-metallic strip type.
Aside from these differences petrol and diesel tank sender are different.
The diesel sender has a low fuel level warning position that activates a warning lamp on the dash panel. It is more important NOT to run out of fuel with a diesel engine, as you have to bleed all the air out of the fuel system. This can be a tedious job on a dark wet night at the side of the road with no spanners.

This shows the later 2A senders, petrol and diesel.
These senders are readily available from parts suppliers from around £15 plus post. You’ll also find that “generic” senders will possibly fit the tank and “work” the gauge. Unfortunately the early sender is no longer made, although you will find NOS from the specialist suppliers, but at a price. It is certainly worth trying to fix the older senders.
Unfortunately I only have detailed info for the later sender, but the principles should be same. The resistance readings could well be different.
Fixing the Fuel SenderHaving checked all the obvious external electrical connections (remembering that the tank needs a good earthing connection to the chassis) and that the voltage stabiliser is working, the only conclusion left is that the tank sender is faulty. You should have a resistance reading of about 260 ohms full and about 24 ohms when empty.

This picture shows the 2 terminal diesel sender unit. There are tabs round the side which hold the 2 halves together, prise these open. It will be impossible to separate the halves until one of the terminal connector strips (E) has been de-rivetted. Before dismantling a cleanup of the unit will help keep things clean and tidy, a Dremil mini drill with wire wheel is very good at removing surface corrosion. The petrol sender is basically the same, without the parts in the RH section and the extra wiper arm (B) which activates the low fuel lamp on the dash.
(Although the picture shows the later type sender, the principles for repairing are the same for the earlier types)
Be careful not to loose the float arm (G) tensioning spring - seen stored in the section on the right. In this unit there are 2 wipers, one for each function - low fuel and fuel level. The low fuel wiper (B) is intact and OK. It makes contact with wiper board C and 'earths out' at position C, switching on the low fuel warning lamp. The fuel level wiper should be below arm B and is missing. This will have to be replaced onto the float arm. The fuel level wiper board has been damaged at point D. This will have to be re-wired. You will need approx 3.5 metres of resistance wire to give a winding of around 260 ohms resistance. Ordinary wire will not do, as it has been designed to have as low a resistance as possible! You need wire that has a resistance of about 80 ohms/metre and a wire diameter of 0.2 mm. Once wound onto the former you may have to remove the insulation/varnish in an arc under the wiper arm, to allow for electrical contact! If you are lucky only one or two turns of wire may be broken and they could simply be soldered together again, else it's a re-wire. To remove this board you will have to drill out the rivets at the positions of A. One end of the wiper board (live) connects to terminal F the other end is earthed through the wiper arm and the body of the sender.
However, the wiper arm itself could have fatigued and fallen off, as is the case here. You will have to make a new wiper. Most model shops sell lengths of springy brass strip for a few pence. Cut & fettled to size a new wiper can be soldered to the float arm (G), you will have to judge the right amount of tension to give contact without wearing the resistance board!
On assembly the case can be closed by bending back the tabs and soldering up the rivet heads at points A.
You may find this link useful
HEREOne thing though – all this sits in petrol, or worse vapour – so why doesn’t the tank blow up with all this electricity connected to the sender

:-X