I wasn't aware that an engine would run without constant electricity being supplied from the alternator, but I will try this.

 

I am not there to help you, but, if you can contact others in your area, you will find someone to help you. You are not qualified to make the correct diagnosis and need someone that is. I am sure some Mini guys could help you there!
I doubt very much it is the pulley, or the belt. The cam chain tensioner and the guides seem like the appropriate items to check.
It is not always the easy stuff is wrong with an engine. Get help! Mini guys will often meet with you, in your area.

 

Though I've never tried it on Scarlet, I would imagine it would start and run until the battery died. I've done it on many other cars. I wouldn't recommend doing it for a long period of time, just long enough to hear if the whine has stopped.

 

Be careful, this is an easy way to fry the alternator!

 

Question: You said that you don't mind breaking out the wrenches to figure out your problem, Do you have the wrenches that you need to break out? Not trying to be funny, but do you have the proper tools, skill set and desire to break your car down to find out what is wrong and the ability to fix it. Wayne brings up a few valid points, none of us are there to help you diagnose your car. When I got my first Mini, I didn't know much about them; however, in one of my previous lives, I built engines and worked as a mechanic. I had a very modest tool set which pretty much lived in a 5 gallon bucket. When I needed to get to a tool, I emptied the bucket on the floor, (seriously). Since that time, I've spent hundreds, if not more making sure I have the correct tools to work on Scarlet. There is a ton of info on this site and plenty of people to give their opinions on what they may think is wrong. At the end of the day, you have to get your hands dirty, ( get some gloves ) and apply some of this knowledge. Have some fun and I hope you figure out what the problem is soon.....

 

I always buy tools as I need them and add them to my collection. And I am no stranger to poking around various engine bays, I've done an entire drivetrain swap on a 1994 a4 Avante with a buddy. My main limitation right now is that I moved to SoCal, away from all my friends and their fancy garages, all I have is an assigned parking space (which I have done reasonably involved work on a Rav4 with). So without the freedom of a garage, I have to be realistic about what I can accomplish.

I have taken this car to a local Mini Cooper independent shop, a nice mechanic from there spent about 25 minutes looking at my car and told me that it was fine, so I went for another opinion from a local performance car shop that has experience with r53s, and they told me the exact same thing. Both of them told me to drive the **** out of it until it breaks and to throw as little money at it as possible lol.

Anyway, new development as of today, I noticed that on the drive from work to a food joint, the noise went away for the most part, hooray! Now it only makes the expected supercharger whine under engine load. Interestingly, the EML did come on during my trip back to the office, but no CEL to accompany it, no performance issues, and when I shut it off and then turned it back on, the EML was gone.

The only other curiosity is that since the thermostat housing (and presumable the thermostat, I'll have to look at my invoice) were swapped, the engine now operates at a baseline temperature of 188f according to my scanguage, as opposed to the 195f that it used to run at before the servicing. Im guessing that the wrong spec thermostat was installed, is this any cause for concern?

 

I still say alternator whine. Listen to these two videos.

 

Did you ever figure out what the whining sound was on your Mini? I listened in your video and mine is the same sound. Would love to know. Thanks

 

I just stumbled upon this revived thread today.
After listening to the OP's video - The described "non associated SC" whine sounds VERY similar to the sound my failing idler pulled made a few days before it's bearing exploded.