Here are some thoughts:

Does it sound like a diesel during the rough idle episodes? It sounds a lot like the timing chain tensioner issues with the earlier N14 models. But that doesn't seem very common on your N18 engine.

It also 'sounds like' a high pressure fuel pump issue, but your 2013 model has the latest HPFP so I don't think that is the cause.

Perhaps it is a thermostat issue? Maybe the temp sensor is sending false temp readings to the ECU and not activating the cold start parameters...

I have a classic mini with carbs but....

my VW GTI with FI needed a cold start enrichment valve replacement to solve a problem that sounds like yours. Very simple if that is what it is

I have the same problem on my 2009 Clubman with 200000 miles. Only mine doesn't go away. So far I've changed the spark plugs, the coils, valve cover gaskets and seals, and the VVT sensors and I still have the problem. The coolant temperature sensor is getting looked at next. You can check your VVT sensors out of the car. Go to youtube and look up your problem. Lots of good videos about Multiple cylinder misfire there. Good luck.

2013 Mini Cooper Countryman S.   On a cold start, the engine experiences a harsh rough stumble idle, it will log Misfires on all 4 cylinders, but more importantly after about 2 minutes, it clears up and no issues until "next cold start" or sitting for at least 8 hours.  It seems to be a timed event of about 2 minutes.  It does not always log the misfires and mini of course says its needs a new $13,000 engine with the existing about 110,000 miles. I can drive this car 90 miles each way to work with no issues, same performance as expected. Its this crazy Cold Start rough idle issue that magically goes away after 2 minutes every time on a cold start.

I have tried the following - New Coil Packs, New Plugs, no change.  Again, its like a timed event Cold Start rough idle like its extremely missing for about 2 minutes and then its normal.