Your Mini Cooper is equipped with an electro-hydraulic steering system. The pump is electric, and moves the fluid through valves depending on steering wheel movement. This system will shut down when there is a system fault, switching the system to manual steer- no power assist.
In most cases, the problem is one of the system overheating. If the system runs too hot, the steering system will shut down. When it cools down, the system will function again. There's actually a recall that was issued by the manufacturer outlining this concern. Here's a copy of the recall:
Campaign - Power Steering Fan Relay Wiring Modification
SI M32 03 04
Steering and Wheel Alignment
This Service Information bulletin supersedes S.I. M32 03 04 dated December 2004.
PERFORM THE PROCEDURE OUTLINED IN THIS SERVICE INFORMATION ON ALL
AFFECTED VEHICLES THE NEXT TIME THEY ARE IN THE SHOP FOR MAINTENANCE OR REPAIRS.
[NEW]designates changes to this revision
Service Action: Correct the Wiring for the Power Steering Fan Relay
R50 (Cooper) and R53 (Cooper S) equipped with Park Distance Control (option 508)
Due to an improperly fitted electrical connection, the cooling fan for the power steering pump may not operate. This in conjunction with certain conditions, high ambient temperatures combined with aggressive driving conditions, could activate the thermal cutout feature causing the electro hydraulic power steering pump to switch off. If this were to occur, the steering system would revert to manual steering without power assist.
The dealership should fix this concern for free under this campaign. If you don't want to go to the dealership, I can post a repair procedure for you. You will still need to obtain the repair parts from the dealership, though.
Let me know if you have any more questions!