All electrical power failed on 2006 Mini Cooper S

- overheating sporadically
- died completely without any warning
- traveling normally at 35mph

2 answers
You need to have it looked at by a trained professional. There are too many potential causes and side effects of such an issues to try and diagnosing it without a proper fault scan and visual inspection. The only obvious thing you can check for is that the belt is still on the side of the engine.
I agree with BPG. Due to the loss of belt you will lose cooling and charging.

Zee