What is the difference between a r50 and r53 mini

The door locks in the R53, R50, and R52 are controlled by the BC1 control module. Unless someone has changed the wiring that would be your #1 suspect.

You can try to change the settings in the BC1 if you have the BMW program NCS Expert or the BMW Coding tool available. Unfortunately, water sometimes drains down into that specific module and causes issues. Other than that, you're left at the mercy of the stealership. Best of luck.

Which model of Cooper do you have? R53?

F27 - 15A

the seats of R53 are fully compatible with R50

You have 3 connectors under the seats:
Yellow: lateral airbag
Blue: seat belt tensioner
Black: the heating seats

the Yellow & and blue must be connected to avoid the errors onto the dash

mini cooper s heat not working - North American Motoring

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www.motoringalliance.com > ... > 1st Generation MINI > Cooper S

low speed fans go bad on all the R50 - R53's. It's very common. The resistors go bad and they're built into the fan . The whole unit will be replaced. You can get them online for a pretty good price, quite a bit cheaper than mini sells them. It's a bit of a job to replace, it takes a couple hours. You have to pull the bumper off,etc.

Step by step instructions and photos
R50 Water Pump Replacement
R53 Water Pump Replacement

Got a Mini Cooper JCW R53, changed the battery on the car and since then have not been able to lock/alarm the car with the remote key. The doors all work and boot with the key inserted into the drivers door. Any ideas what the cause is?

It's the rear parcel window shelf.
If you want a new one, you can buy it from: http://www.realoem.com/bmw/select.do just fill out the information (R50 if you have a Cooper, R53 if you have a Cooper S) then do a search on shelf (you will need to buy shelf and cords)
If you don't mind a used shelf, you can search on your local MINI forums, northamericanmotoring.com or ebay.