I'd log the temps being reported by the coolant sensors and fuel temp sensors, from the moment the engine is first started cold before any oil/coolant/fuel has had a chance to warm up - to see if the values look within normal ranges.
When you first start the engine in the morning you'd expect them all to report temps somewhere near outside ambient temp (but will start to change fairly quickly, which is why you'd want to log right from startup).
If any of these sensors were reporting the engine being hot when it's actually cold (but still within allowable ranges so the ECU doesn't throw a fault), this could cause overfuelling that's worse when the engine is cold, but less noticeable when it's warm.
VAG coolant temp sensors are renowned for failing.