[SpYDd]

Just an update on this, car went back into the Indy to diagnose.
I asked if they did the relearn procedure and they did but after a few days it seems it was forgotten by the car.

They found some corrosion on the Valvetronic and DME harness (and DME itself).
They cleaned off the corrosion and did all the relearn in ISTA etc.

The car now drives great, no jerky gear change and no hesitation or flat spots at all and my mpg’s have increased greatly. The car feels smooth and it’s quiet when idling. I’ve been driving it about a week now and the difference is night and day.

However I’m still getting the rough idle and shaking on a cold start.
It doesn’t seem to do it every time but when it does the symptoms are a slightly longer than usual crank, rough idle for a few seconds until it settles down to normal idle speed and what feels like a side to side shaking or wobbling that can be felt by driver and passenger.
All the symptoms go away after a few seconds and the car drives and starts fine for the rest of the day. After sitting overnight the symptoms return.
I tested the other day by pressing the start button without my foot on the brake, wait about 1 minute and then it starts normally it seems.

Interestingly we discovered the car has a DME from an e60 in it.
I had a fault code for the Ebox fan that was cleared by cleaning the corrosion.

Do you think the symptoms I am having are related? Or something else?
Guessing the next step is to replace the DME and harness.