So the code you have there is for the VL (variable valve lift) system. You must have a CESA/CETA engine--EA888 Gen 2, would be 211hp/258lb*ft rated.
I dunno what kind of thing these guys are smoking with this 2000 pound estimate here, but it must be some expensive stuff if they need to charge that much to support their habit.
I get the feeling that kind of "ballpark quote" is more what we call an "eff off quote" here--i.e. meaning they don't want to deal with you or your issue, and would sooner just send you elsewhere because they know you're not going to pay 2000 to repair it. But I mean they kind of already admitted they don't want to deal with it so the 2k is a nothing number they just randomly gave you for that same purpose.
P11A1 is the OBD-generic code for "cam shift actuator" as you say, though I don't believe VAG calls it that. Essentially "cam shift actuator" is what is changing the lift profiles of the exhaust cam. There's two solenoids per cylinder IIRC, located on the back of the valve cover (or more correctly, cam cradle, on EA888s).
Since the car is behaving rather violently though I would highly recommend
not driving it until fixed--you could make one problem a whole lot worse when we're talking about a potential camshaft issue. It's one thing to have the cam profiles not locking in correctly, it's another thing if you lose cam timing or have some valve issue that then causes valves to hit pistons--at
that point you're probably going to be looking at "a new engine" (or more correctly a cylinder head rebuild, at a minimum). That kind of thing is not
supposed to happen (read below about what happens during an actuator fault), but that doesn't mean it's not possible when we're talking about a camshaft issue.
You may have a look at this thread for someone that had an A4 with this problem:
A4 (B8 Platform) Discussion - Help with Engine Code P11A100.. - Hello, Below are the details Make: Audi Model: Year: 2013 VIN: WAUZZZ8K5DA160377 ODO: 127040 km The car is showing open circuits on all the cam actuators.. Please help with the specs: 1. Ignition on : I am getting 2.5V on the...
www.audiworld.com
This is on a longitudinal car (TT is transverse) and a 1.8T (apparently in South Africa they had a VL 1.8T??), but it talks about the same code. However some of it may be more technical than helps you out, which I understand.
Indeed you may not exactly want to "become your own mechanic", I still suggest giving this a read since I'm sure you're capable of reading through and at least getting a gist how this system works in general--
pages 5-13 of this document (below) outline the workings of the AVS (Audi ValveLift System) which is where your problem is:
Don't let "solenoids" scare you, they are just electromagnetic switches which are either in one position or another, controlled by electricity.
IIRC during most failures of the system, the engine should still run relatively well, though not correctly, and you will get the MIL (Check Engine Light) coming on. As we can see here:
Audi SSP 922903 said:
If one or more actuators fails, the ECM will initially
attempt (several times) to change over to the other cam.
If no adjustment is made, the cam elements that cannot
be adjusted remain in position.
All other cam elements are changed over to the large
cam. They then remain in this position while the engine
is running. The next time the engine is started, another
attempt is made to adjust all cam elements.
The long and the short is, it has to do with the actuator solenoids on the VL system. I don't think this is a 2000 GBP problem TBH. It
could be, but it doesn't seem like anyone has actually
diagnosed the problem to know--you've just been getting eff off "quotes" from places instead of an actual diagnosis, which is unforutnate.
Likely, either the solenoids themselves are the issue or the wiring/fuses/circuits related to them. A quick(er) check of the solenoids is to swap the ones indicated as problematic (presuming "A" is cylinder 1?) with the ones on another cylinder and then see if the code follows the solenoid(s) when it reappears (presumably B, C or whatever they've been moved to). If it
doesn't, and remains there (on "A") you know the problem is either electrical/wiring on that cylinder or it's something mechanical at that cam lobe.
The latter (mechanical situation) may indeed cost a lot of money to fix (e.g. a new cam); but something related to the electricals here should cost nowhere near that kind of money. Verifying that stuff electrically does not requre any kind of "special samurai wrench". I suspect this is not "too big a job" for any of the places you've taken it to, it's just out of their expertise diagnositcally--IMO that means they shouldn't be in the business of fixing cars, but that's another story.
First thing's first you need a proper diagnosis and estimate after someone has actually dug into it and told you what's needed to fix it, not this "oh it's too big a job" or "it's 2000 pounds, go away now", etc. type of thing that you seem to be getting.
I mean how "big" a job is it, if you even needed a new exhaust cam, say? If
that's what they call a "big job", I think these guys should close up shop and call it a day. I only see this being 2k+ pounds if you needed a new/rebuilt cylinder head (small possibility but I doubt it), and perhaps they'd call
that a "big job" but diagnosis on the VL system and telling you what needs to be done
from there, that's just standard work. After that,
then you'll know whether you're into simple things like electrical repair, new solenoids; or something more costly like a cam or entire head.