This is a guide for those like me who end up in this situation of having to bleed the ABS pump, not those just doing a normal brake bleed. However this procedure below will also bleed your calipers and do a full fluid change at the same time anyway (approx 3L of fluid).
I was changing my brake fluid which "usually" takes about 1ltr of fluid if you're just doing the calipers. However, I wasn't paying attention to the reservoir levels as I did it and managed to run low on fluid and suck air into the brake system. I bleed a little air out of the front calipers and everything seemed OK afterwards. The pedal felt normal for day to day "A6 rather than RS6" style driving but once you did three or four 70mph-10mph stops in a row the pedal was extremely soft and I couldn't get the ABS to engage at all.
At that point I realised I probably had sucked air into the ABS pump during my earlier bleeding so I scoured the internet which was filled with mostly bad or incomplete info, rang Unit20 and a few others etc and no-one it seemed had any solid "this is how you bleed the ABS" info for me so, following what bits I picked up off the internet, and then working out a few things myself on the car using Elsawin (also not the complete info), here's what is actually needed to bleed the ABS pump on a car with a Bosch ESP 8.0 system (full name for search engines to pick up on) and it works.
First off you'll need about 3L of fluid (4L is better to allow for some waste but 3L will juuuust about do it). I chose Castrol React SRF racing Fluid, as it has the highest "wet" temperature boiling point by a huge margin over any other (wet temp is after it starts to absorb moisture over time). It's not cheap though but you can find it on Amazon for £50 for 1L.
Note below that the procedure from Audi says you need a minimum of 2Bar pressure if using an eezi-bleed etc following this procedure and NOTE THE CALIPER SEQUENCE -
So, bleeding the ABS pump itself! You'll need VCDS and either two people (preferred) or one person with a brake pedal holder/stick as the brake pedal will need continually pressing whilst opening/closing the caliper nipples during the procedure.
Starting with the caliper sequence as noted above (yes ABS also does the rear calipers too), hook up your waste fluid bottle to the inner nipple of the Front Left Caliper then open VCDS, Select Module 03 ABS Brakes and then, VERY IMPORTANT (and the bit missed out of most online versions), go to Security Access and type 40168. If you do NOT do this you can NOT run any ABS pump tests!!!!!!
Now you have entered you security access code you can select Output Tests which will automatically start the ABS test procedure and the ABS light on the dash should start flashing and you'll hear an electrical pump sound. At this point VCDS will guide you through the rest of the procedure but effectively you do the following to each caliper -
On VCDS press "Start/Begin"
Lightly press and hold the Brake pedal and DO NOT LIFT the pedal until you are told to stop - it will take a few minutes per caliper and lifting the pedal early will suck in air!
On VCDS press "NEXT" which actually starts the test - you'll feel the pedal shake/pulse and hear a new pump sound
Open the caliper nipple - fluid/air will now leave the caliper
Wait until about 500ml of fluid has left the caliper
Close caliper nipple
Connect waste fluid bottle to outer caliper nipple
Open the caliper nipple - fluid/air will now leave the caliper
Wait until about 500ml of fluid has left the caliper
Close caliper nipple
On VCDS press "NEXT" - the pedal shake will stop
Take your foot off the brake pedal
Move to next Caliper and start again (don't have to press start though, follow the instructions in VCDS)
Repeat the same procedure (brake on - open - bleed 500ml - close - brake off) for every caliper in sequence, getting about 500ml out of each which is exactly 3L by the end.
Once all the calipers are done you can press "Done/Go Back" on VCDS to finish the procedure. Make sure every nipple is tight, take your car out for a drive and try to get the ABS to activate on gravel, dirt, mud etc and make sure it pulses the brake pedal and you're all done.
Again, this is only for those who want to either flush the entire 3L of fluid out of the brake system (calipers/pipes together hold 1L whilst master cylinder/ABS pump together hold another 1.5L) or need to bleed air out of the ABS pump.
Hope that helps!