Did you check the coolant level? May sound silly but if the fan works of a negative temperature coefficient sensor it will be positioned relatively high up on the radiator and if the coolant level is low it's unable to read the temerature of the fluid, and unable to earth the relay to switch the fan on.
If it's read from an engine sensor and switched on by the onboard CPU (the cars brain) then that sensor has failed.
You also could check the mechanical thermostat (it'll be mounted probably to the rear upper side of the engine and easily recognised by having at least two coolant pipes running into it) That'll require you to drain coolant, not that hard.
Put the thermostat into boiling water. If it opens it's working, if it fails to open it needs replaced.
These are the easiest options, and the fault should be repairable by and half decent garage instead of an Audi dealer.
If it's an electrical issue use an auto spark.
Good luck.