No idea on the screeching, I don't think that there are any belts, so it would likely be one motor; maybe the heater blower? Is this starting up the car, or starting it moving? If only while moving, that implies something in the propulsion, if when turning the car on, one of the few things that starts up with a motor then is either the heat pump, a/c compressor, power steering pump, or the blower motor.
BMW (and many other cars) require/suggest that you flush the brake fluid on a 2-year interval based on the date of manufacture (not the date of sale). It's probably time for that service. It should not be ignored. Brake fluid absorbs moisture even through the rubber hoses...this can lead to corrosion from the inside, and also affect braking performance if not serviced.
2011 535i x-drive GT, 2014 i3 BEV