Run through disassembly, clean and rebuild of the electric power steering pump for BMW mini R50 and R53 (2000-2006). I do not cover removal and replacement. That is already well covered in other videos like modmini
If dust creates a short then the pump can run all the time. Conversely, if the alternator is not working then a good power steering pump will not run at all. This is a logic problem - if the charge warning light (on speedo) is illuminated, then the power steering pump will not be allowed to run. Actually, the yellow wire to the dash light itself also goes to the PS pump for this logic loop. A bad battery or alternator can cause loss of power steering, even if the car runs. To diagnose this - look to see if dash charge light flickers on at times when PS fails.
Another logic constraint is the ignition needs to be on (fuse F39 5A in foot well, green white wire). Power to the pump is direct from the battery (heavy red cable, fuse FL4 100A). Switching and control is from a module within the pump itself. Violet-white wire is the signal wire from the pump to the ECU, ABS and OBD.
Some numbers for the electronic parts:
Smaller transistors closest to power cable connector - G76 and C67 (= 2N04H4 TO 220 ).
These fail causing always or never running
Capacitors (all the same) 2700uF (M) 25V epcos LL B41866-S5278-M5 40/125/56
Big rectifier diode schottky PBYR 4045WT PHm0419 A2
Diode (ON) MR2520L
Small capacitor 47 VFC 6M1
Fuse F39 5A in left foot well fuse box (location map inside of cover)
Fuse FL4 100A in engine base fuse box UNDERSIDE (remove air box, unbolt fuse box, roll over to access underside)
Bearing - @jsse commented it is a 6001 2RS