I have an 09 MCS Hardtop w/ approx 8k miles and have had the same problem a few times over the past few weeks.

The first time was immediately after the first heavy rain of the season, so I thought maybe some water leaked into the cylinders.

However, it has now happened at least twice in dry weather.

Giving it some throttle for about 30 seconds seems to level out the issue until the engine warms up.

I will definitely try to catch this on video, but that will be hard to do since it only occurs randomly.

 

I think the HPFP is likely. Have you ever had to push the start button several times to get it to start first try in the morning? I see it's cold where you are but that was my first tipoff for my fuel pump. With mine once it started it ran just fine and started fine afterwards. But it wasn't related to cold.

 

Sigh...
Took my 2012 MCS (8500Km so far) in on Monday for a similar problem - stumbled a couple of times when cold and wet out - somewhat random occurence - and after a minute or less of running all is OK.
No CELs or fault codes logged.
On Friday, the problem got worse - 4 presses of the start button, where it would crank, catch for a split second, and then die. Finally, on the 4th attempt, I had to give the engine a bit of 'gas' to keep it running - the equivalent of trying to run an unloaded engine at 1500RPM or so. After less than 30 seconds of operation, the problem vanished again! Still no CEL or codes.
The only common points were that the outside air temp was between 0C and 5C each time, and it was wet out - either damp or raining, after an overnight or longer of sitting. My car is parked outside - the garage is too full of reno debris and parts these days! :(
I don't suspect the fuel itself, as I have used Esso, Petro Canada, Shell, and Costco with no difference - in all cases 91 octane, and the stumble occured for sure with Esso, Costco, and Petro Canada fuels.
I have also noted that on start up, after a onvernight of sitting, the engine coolant temp does not match the outside air temp - I have an EcoRoute HD and a Garmin GPS to display my guages, and have noted that the coolant temp is usually 20 or more degrees C higher than the engine air intake temperature. My service advisor did add that point to the fault description given to the tech...

MINI Markham has had 'Summons' for 2 days so far - the dealership reports back regularly with "no concrete findings so far. Keep the rental, and we will keep trouble shooting..."

It is interesting to see the fuel pump come up repeatedly in this thread as the fix - I will report on what MINI does to my MCS - I'll suffer the Fiesta in the mean time... :( :( :(
E