Fighting with 1125 and/or 1126 codes. What already is done:
1) TB is replaced by the brand new OEM VDO unit
2) Pins of ECU and wiring loom (X6000) cleaned even though those looked cleaned from the beginning
3) Wiring checked by multimeter (from ECU to TB connector, all 6 pins were checked). Of course, wiring still can be an issue, even if it seems ok from the multimeter point of view

The issue may come and go. Last time, I've re-sited the ECU X6000 connector - the codes disappeared and the car drove for some time, until it threw these codes again. This time I re-sited the connected at TB - and the code went away again. Which makes less and less sense.

I know, MINI had a SIB describing the similar issue where they recommend to replace the engine wiring loom. Does anyone have the PDF of this SIB? I can't find it anywhere :(

Another theory - ECU. But that is even more major PITA than replacing the wiring loom (which is already unavailable from the dealer, at least for 2006 Cooper S Auto).

 

Some updates. TL;DR; I really doubt it is TB or even wiring issue.
DR; I now have 3 throttle bodies: the first is my original one (MFD 2006), another one is used one from ebay (MFD 2007) and finally a currently installed, brand new (old new stock), VDO, MFD 2019. I've measured them thoroughly with a multi-meter:
NEW TB. Was measured through the wiring at ECU connector. Pins here are # of pins at TB connector, not ECU's ones.
Pins 3-5 4.1 (here and further - Ohm)
Pins 1-2 1500 -> 530 (fully closed to fully opened)
Pins 4-2 600 -> 1300
Pins 6-2 995
Pins 6-4 1250 -> 424
Pins 6-1 850 -> 1429
Pins 1-4 1811 -> 1760

TB from ebay (measured at the TB connector):
Pins 3-5 4.2
Pins 1-2 1328 -> 457
Pins 4-2 542 -> 1201
Pins 6-2 905
Pins 6-4 1143 -> 392
Pins 6-1 739 -> 1275
Pins 1-4 1616 -> 1572

My original TB (measured at the TB connector):
Pins 3-5 5.1
Pins 1-2 1374 -> 451
Pins 4-2 570 -> 1200
Pins 6-2 904
Pins 6-4 1169 -> 386
Pins 6-1 789 -> 1268
Pins 1-4 1692 -> 1565

So, basically, all of them are fine. I was testing wiring as well (as I mentioned I was testing the current TB through the wiring loom, I also wiggled both ends of the wiring loom (one - from the TB to the 'spider body', and another - from the 'spider body' to ECU). No issues found so far. I have another idea to test it with the oscilloscope and a test signal (like 1 kHz) and see if signal is affected during wiggling of the wires. But I really doubt it, tbh.

So, my main theory now is ECU. Going to send it to check it up. Any recommendations of the ECU services?

 

Try to clean with WD40 the throttle body female connector from wiring side.Clean also the TB male connector.With a sharp needle try to close it a little each female connector.Is common issue at these cars after 18-20 years ,the female connectors fail.Can try another repair by replacing full female connector of the Throttle Body.

 

Yeah, I guess, I'll de-pin the socket and either re-pin it or at least adjust them by a pin. Worth trying.

 

Ok. Re-pinned and replaced the TB connector. Brand new pins (p/n 61136820598, 6 pieces), brand new seals (p/n 12527503411, 6 pieces) and brand new connector housing (p/n 12527505669, 1 piece). All bought in my local BMW dealership for speed (and it is much cheaper than same parts from my local MINI dealership!).

Will see how it goes. So far, so good (after 100 miles of driving)