the devil and the deep blue sea ,,, best case its a wire rubbed through and have a positive to earth fault with that line from the dme worst case is the dme ie ecu is faulty cheapest way to fix ecu issue if it is , is to buy a ecu and key and bcm kit off ebay link below for a idea
main dealer will ask £700 for ecu plus programming extra £200 plus a £190 for diagnostics,
another way is to find a ecu have it cloned £250 ish with ecu,
if engine harness link below
the coolant temp sensor is run via engine ecu is switch earthed via ecu

Mini One R50 00-06 1.6 16v Petrol Manual ECU kit key 7515130 7513981 6918352 | eBay
Engine ECU: 7515130 + 7513981. Clocks: 6918352. Model:One R50. Valves: 16v. Fuse box: 1509011 518030317. Fuse box: 1480790. 1 x ECU Kit. Make: Mini.
www.ebay.co.uk
another way is to find a ecu have it cloned £250 ish with ecu,
if engine harness link below

Mini Cooper One R50 Petrol Engine Wiring Loom Harness | eBay
Find many great new & used options and get the best deals for Mini Cooper One R50 Petrol Engine Wiring Loom Harness at the best online prices at eBay! Free delivery for many products!
www.ebay.co.uk