https://www.ebay.co.uk/itm/AUDI-A8-4...-/332302365216
4E0121113 This is the usual culprit, Audi call it a Coolant Regulator.
I don't know about 4.2TDI location though but seems to be present on many VAG models, from Golfs to Bentleys, including our A8 3.0 TDI, W12 and others. Usually it's beside the gearbox where the coolant pipes enter it or behind the radiator near the bottom.
When it fails, coolant can bypass the main thermostat and run through the radiator and autobox even when cold.