MINI's have a common water reservoir and washer pump for the front and rear washers located behind the plastic wheel arch liner in the void below where you fill the washer bottle up on the drivers side (UK). Look at the black plastic inner wheel arch liner and you will see a mixture of screws and plastic retainers - access is much easier if you remove the front wheel (or use a cross headed stubby screw driver with the wheels on full lock if you prefer to struggle) - take them out and put them to one side, remove the front fixing underneath the plastic side skirt sill as this also retains the bottom of the arch liner. Having removed these you can now bend/twist the complete liner off the car or leave the front part in situ and flex the rear part away to gain access as required.
The pump is bi-directional, one direction pumps water out to the front windscreen washer jets, the other direction pumps water out to the rear washer jet. The change of pump direction (rotation) is simply controlled by changing the polarity of the voltage supplied to the single washer pump motor.
If the motor does not operate in one direction it could be a relay fault.
The polarity changing is carried out by two relays. These are hidden from view and are not very accessible. To access the relays you have to open the passenger door and partially remove the rubber door sealing gasket that runs along the door frame - where it changes direction from the door sill to the vertical part of the door frame. Remove a single plastic push fit holder to the upper right of the fuse box access lid. Use some gentle force to push back the plastic kick plate around the fuse box. Standing outside the passenger side of the door looking in as you push the plastic kick plate back you will see the two 'light grey' relays that perform the polarity changing. The relays are identical and therefore interchangeable. If you swap the relays over and the fault transfers from say the front to rear washers or vice versa you will know the relay is faulty.
Part number for the relay is 6136 - 1393415 (last 7 digits are the actual BMW part number) Approx. Price - £6.50. Re-assembly is reverse being careful not to pinch the wiring harness.
If the pump makes a noise in both directions but there is no water and the pipes/jets seem clear, check for water at the pump end with the outlet pipes removed, this should indicate whether it is a pipe blockage or pump fault. Also check the rubber sealing washer and filter combined are clear by unclipping the pump from the plastic reservoir bottle. My pump failed a while back, it was whirring but not pumping any water and required replacement due to a broken plastic drive shaft inside.
Washer Pump Fuse is in Passenger compartment fusebox F23 20A
Mini's fitted with the headlamp washer option have another water reservoir/washer pump located on the opposite side of the car in the gap between the passenger footwell and outside of the car. Access is the same as above by removing the opposite plastic inner wheel arch lining.
Windscreen washer - weak flow:
Photo of rear of fuse box (borrowed - not mine!) which I think shows the two 'grey' relays for washers (to be confirmed anyone?)
The plastic base of the rear wiper arm hinges up to often reveal a blocked and rusty water nozzle underneath.