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 later ones (5000, 100, 200 from about 1984, 80, 90 from 1988 on) use a pump/controller unit with a seven pin connector (of which only five are used). Here is an electrical and pressure line schematic of the dual control system:
Remember that since the control switches are in the front doors, and the pump is buried in the trunk, 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 front doors, 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 switches in the front door actuators, 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 front doors, and digging out 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 multi 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, becomes three wires at a welded connection hidden somewhere in the loom, and then into both front doors and to the pump connector. Yup, it's time to remove those door panels, if they are not already off.
The door wires go all the way to pin 1 on each master vacuum actuator switch connector. Remove the connectors. Use your voltmeter to check for 12v at terminal 1 of each of these. 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.
The third of these red/black wires goes to pin 1 of the pump/control unit connector. Undo it as well and test in a similar fashion. Also, wiht the battery disconnected, measure the resistance between all three of these "pin #1"s and ground. It should be zero between the pins and infinite to ground.
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 switches in the actuator have two more wires. Pin 3 is a brown wire and should exhibit zero ohms to the chassis ground of the car. Pin 2 is either green/blue or green/red, depending on the side, and runs to pins 3 and 4, respectively, of the pump unit. Check the resistance of those wires, from door connector to pump connector, as well.
If you do have problems with the wiring, the most likely location for breaks/shorts is in the door jamb boots, but they could be elsewhere along the way if you are really unlucky. If you are going into a 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.
Testing the pump is moderately complicated, since it has so many connections. Pin one should get 12 volts continuously, and pin 2 should be grounded. Pin 6 is connected tot he ignition switch, but I do not know at this time whether it is to prevent operation while starting the car, or some other function.
Pins 3 and 4 are sent either a ground or 12v signal by the actuators, depending on their position. Apparently the pump operates when they are different, in the direction commanded by the most recent one to change. Got that? To test it, you would have to apply 12v to both, then switch one to ground. The reverse also applies - ground both and switch one to 12v. It may only be necessary to apply a signal and change to one of them, I do not know for sure.
It could be that the best way to test the pump is to ensure that all the electrical function preceding it are working properly, and then plug it in and try it out (see below) - but this risks hurting your wiring if the pump draws too much current for some reason.
Anyway, it should alternately cycle as a pressure and vacuum pump, running for about 30 seconds in each mode. 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 doors are wired properly, and the fuse is good and delivering 12v to the master actuators, you can further test them by measuring the voltage at the connector for the pump unit. Pins 3 and 4 should each register 12v or ground depending on the position of the appropriate lock knob. 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 front doors, one at a time. 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!