I hate to say it but it looks like it is by design - retarded ignition for the first several seconds until the CATs light off... Here is a quote from an article on Edmunds.com (link follows):
"There are other times when the Mini's is uncharacteristically coarse. It happened when I fired it up first thing in the morning and got underway, and at the first traffic signal 100 yards away as it rolled to halt. That's when I noticed it. It was a thrumminess that I'd not observed previously.
A few seconds later, like a switch, the coarseness stopped. Just like that, it was back to its usual unusually-smooth (for a 3-cyl) self. All while I waited at the red light.
So, what's going on here? First off, it's completely normal. During a cold start modern engines run very retarded ignition timing in order to introduce additional heat into the exhaust stream. More specifically, to help light off the catalyst, which needs to cross a certain threshold temperature before it functions in earnest.
This highly retarded ignition timing results in increased combustion instability during idling conditions. In other words, each combustion chamber's "burn" wants to do its own thing, relatively speaking. There's a wider variation in output among all cylinders, and also a wider variation in each cylinder's output from one burn to the next.
This combustion instability manifests itself to the vehicle's occupants as rougher operation.
Once the engine controller is satisfied that the catalyst is lit off (a decision that involves the input of air temperature sensors, the oxygen sensor, coolant temperature sensor and more), it switches the ignition timing out of cold start mode and reverts to a less-retarded ignition calibration. It's a switchover you notice in the Mini, more so than in some other cars."
https://www.edmunds.com/mini/cooper...014-mini-cooper-temporarily-rough-engine.html