There are basically two systems of pneumatic central locking on Audis of the vintage that I deal with - the earlier cars are controlled by the drivers door lock, and the later ones by both front door locks.
The early ones (5000/100/200 before 1984, 4000/Coupe/Quattro/80/90 before 1988) use a pump/controller unit with a three pin connector. Here is an electrical and pressure line schematic of the single control system:
Remember that since the control switch is in the drivers door and the pump is buried up by the passenger side trunk hinge, the wires are a lot longer than this picture might make it look. Also, the pneumatic lines snake all over the car, under the carpet, and into each door (and the trunk lid) to operate the actuators.
The first step in troubleshooting is to determine whether your troubles lie in the electrical or the pneumatic side of the system (or both...). If you can hear the pump humming away when you lock and unlock the drivers door, but nothing happens to the other door locks, it is most likely related to the hoses connecting all the lock mechanisms, the pneumatic actuators in the doors, or the pump no longer creating effective pressure. If you hear no tell tale sounds, then the first place to look will be the electrical portion of the system - the switch in the drivers door actuator, the fuse and wiring, and the actual response of the pump to appropriate signals.
Location of parts:
I think that apart from the most cursory checking (like, is the fuse there?), you can count on opening up the drivers door and digging under the trunk liner to get to the control unit/pump. If holding your trunk open while trying to remove the pump is a chore, perhaps replacing those dead trunk struts should be the first order of business? They cost about $20 each and should be replaced, not lived without. What is the point of a luxury feature like power locks when your trunk won't even stay open on its own?
I would recommend doing all this work with at least one window wide open - that way, you can always get back into the car if you somehow manage to lock all the doors so well that you can't unlock them!
When you extricate the pump, you will probably need to unscrew its ground connection to get good access to the 3 pin connector and wiring going to it.
To test the pneumatic lines, disconnect the main feed from the pump and apply vacuum to it to lock the doors, and pressure to unlock them. A Mity-Vac type tool is best for this. If your system fails this step, examine all the pneumatic parts for leaks or disconnections. Most of the lines are a hard green plastic tubing. At the junctions there will be short pieces of fairly durable rubber hose, and Y's, T's and such to connect the branch lines. Check the accumulator thing, which is a bright green sphere or cylinder. Check all the door actuators, which are usually blue oval things, about 3" by 1", with a collapsible boot type thing on the top attached to a metal rod linked to the latch mechanism.
Since it is possible for a very sloppy door handle replacement to have displaced one or more of these rods, make sure they are all in place, too. Simply locking and unlocking the door in question manually, while watching the actuator, will show whether these are intact.
Repair or replace any faulty components in the pneumatic system after doing all the troubleshooting. It is quite possible that at some point, the electrical systems failed, and someone "opened up" a connection to stop the locks from all pushing each other up and down when operated. Finding such a disconnected part may simply indicate that you have other things to fix as well.
If your system exhibits electrical malfunctions, test it the following way.
Start at the beginning - in the fuse box. Fuse 19, 10 Amp, should be getting battery voltage (always on no matter what key position) on its hot side, the fuse should be good, and the battery voltage should then be present on its fused side as well. If the fuse is blown, it may indicate a pump drawing too much current, or a wire shorted to ground somewhere.
If that checks out, make a note that the wire leaves the bottom of the fusebox at M30az, which I think is a single tab connector. You might need to check that, too, but it is unlikely to have fallen off.
A red/black wire then proceeds through the spaghetti mess under your drivers side dashboard and into the door. Yup, it's time to remove that door panel, if it's not already off.
This wire goes all the way to the drivers door lock master vacuum actuator. There will be a three pin connector at the actuator. Undo it. Use your voltmeter to check for 12v at the center (#2) terminal. If it is not there, then this wire is probably broken in the door jamb. Make a note of that. (If your fuse was blown, you should be checking for continuity between the "fused" side of the fuse holder and this terminal - and ground, since a short would also mean continuity to ground. If your fuse was blown, also try measuring that resistance to ground while moving the door around.) The reason I keep saying to "take notes" is that you should troubleshoot the whole system before digging out your repair tools - that way you will know what to be expecting to fix, rather than finding one more thing after another as you go.
Test the actuator switch itself, for continuity between pins 2 & 3 and pins 2 & 1, one should correspond to the locked, and the other to the unlocked, position.
Next, the switch in the actuator has two other wires leaving it. Check the continuity between pin 1 of the door connector and pin 1 of the pump connector (and ground, to check for shorts), ditto for pins #3 on each. There should be no resistance (zero ohms) between the pins, and infinite resistance between all of them and ground.
If you do have problems with the wiring, the most likely location for breaks/shorts is in the drivers door jamb boot, but they could be elsewhere along the way if you are really unlucky. If you are going into that boot to fix breaks, one, remember to replace a section of wire about a foot long to keep your splices in the door and in the car (under the little kick panel - pull up its little bit of carpet to reveal its fixing screw, and undo the hood release handle screws as well) and not in the part that flexes, and two, to look at all the other wires for damage while you're in there. If any of your door controls don't work, their wires may also be hurt. Only cut one wire at a time, to avoid confusion in the splicing process.
You can test the pump by applying 12 volts to pins 1 or 3, one at a time, while grounding the brown wire. It should alternately cycle as a pressure and vacuum pump, running for about 30 seconds in each mode. If the pump works fine, a further test of the wires (green/blue and green/red) from the actuator to the pump connector is to try running 12 volts along them to the pump to make it work, remembering to keep the pump grounded. Definitely do not zip the project up until that pump is consistently performing both its functions. If needed, another pump should go in rather than the one that does not work right.
By now you have probably found one or more differences between "what should be" and "what is," and hopefully you have been taking notes of where the discrepancies lie. Fixing all of them at once is easier than doing them one at a time and still finding that things do not work. That is why you do such an exhaustive troubleshooting process first.
Once the door is wired properly, and the fuse is good and delivering 12v to the master actuator, you can further test it by measuring the voltage to ground at the connector for the pump unit. Pins 1 and 3 should each register 12v in turn, when the master door is locked and unlocked. If your pump seems to be in good working order, plug it back in and ground its brown wire (it may be difficult to attach it to its original location while troubleshooting, so use a jumper wire). See if it cycles properly upon locking/unlocking the drivers door. If it does, plug the (repaired) pneumatic line back into it and test the system again.
When you are certain the system is working 100% (no less!) of the time, then you can replace all the parts where they belong. I recommend testing the system at each stage of reassembly to ensure against accidents - if it suddenly stops working, at least you know you only did one thing since it last worked. Make sure the brown wire from the pump connector is properly grounded as you reasemble everything, at its original location, or a new one of the old one is nasty looking. The sheet metal screw, the ring tab, and the sheet metal of the body should all be shiny and clean, free from dirt and corrosion. Use some dielectric grease on it (and the door and pump connectors) to keep it that way.
Have fun and good luck!