The EHPS cooling fan is activated by the EMS when the engine temp reaches 122C (251F) which is the High Speed fan stage on the Engine cooling fan. Once the Engine Temp drops by 4C (7F) then the High Speed circuit stops and the engine cooling fan then reverts to the Low Speed circuit resulting in the EHPS cooling fan stopping.
The EHPS Cooling fan is not connected to the EHPS pump, the pump will shut down if it reaches a set temp to protect its electronic control circuit.
Hope this helps
Ref: Mini Cooper Diagnosis Without Guesswork, Bentley Publishers.