This isn't Mini specific but usually with these kind of faults you may need to replace the other side sensor, then move onto the reluctor/ABS ring on the hub, then the ABS module/ECU. Basically the ABS ECU doesn't know what the drive wheels are doing as the message isn't getting back to it, so it will shut down the ESP and sometimes the ABS, it doesn't know how fast the wheels are spinning so the speedo shuts down, if you've got cruise control that will have shut down too. ABS modules can be repaired and of this age is quite common.
I've just gone through similar issues with my Volvo, the Saab we had previously also had the same faults. The former was reluctor rings and a sensor the later was a refurbed ABS module.
I expect out 04 Cooper S will do the same within a year or so. It usually happens after or during winter as sensors and connectors corrode.