Is it a manual or CVT (auto)?
From that video, that to me sounds like a supercharger whine which may suggest that you have a pipe loose or airbox cover not on correctly (latter wont affect MPG or driveability though). It could be belt related as well but that tends to be a squeak
If I'm right and its supercharger noise, then if you have the car parked up and in neutral, free rev it up to around 2500-3000 and hold it there (video if it you can) - then 'blip' the throttle to encourage the supercharger to turn on. The supercharger will initially spin up but then should disengage. You can hear the supercharger normally, however if something is wrong, IE airbox not connected properly or other pipe disconnected then it will be louder than normal.
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A quick video of mine just now with the air filter and cover removed just as a comparison for you. I free rev it to 2500-ish, no noise (which may explain why you cant hear it at idle looking under the bonnet), a quick throttle blip whilst maintaining higher than idle revs and you can hear the supercharger whine clearly and then it turns off again.
0-8 seconds - free revving to around 2500 (supercharger disengaged)
9 second mark - quick throttle blip to engage the supercharger
9-12 seconds - supercharger whine
12 seconds onwards - supercharger disengages and noise goes away
