I can figure out the video but Are you having misfires with this? This sounds exactly like when my HPFP was starting to go and not long after I started having misfires extremely bad

Accurately diagnosing any car issues remotely is difficult at best. All we can provide is our best 'guess-timate'.

The symptoms on your 2008 Cooper S you describe 'can be' the high pressure fuel pump starting to fail.

Mini Cooper S Oem High Pressure Fuel Pumps 2007-2010 (minimania.com)

But the similar symptoms can be caused by other issues...
pcv valve / valve cover
vacuum leak
bad gas
defective cam or crank sensor

Are there any fault codes recorded in your MINIs computer?

My 2008 Mini Cooper Sport (R56) has been difficult to start for the last couple months, often taking 3-6 attempts before it manages to stay started (it starts and cuts out after a second or two). When it eventually starts it runs rough,

During the video I gently placed my foot on the gas to bring the car to around 1100 rpm. As you will see, it jumps up to around 2,000 (without me moving my foot on the pedal). I then I took my foot off the gas and put it back to 1100, applinked when it continues to run roughly. I should add that the engine was warm when I made the video (but it happens when the engine is cold too).

I've had my timing chain replaced recently because I was told it was my timing but it made no difference. Any ideas what it could be?