Overheating issue, potentially bad reservoir cap?

Get a sniff test done to see if the head gasket has gone.

My Mini R56 N12 had a similar problem. Having gone through the whole system by testing fuses, relays, thermostat, fan, water pump and the friction drive to the pump I then changed the reservoir cap and everything worked as it should. I still need to check the head gasket to confirm all is well. There are a couple of quick tests you can do:
Check the fuses for the fan (engine fuse panel)
Bleed the cooling system - use the bleed point on the thermostat housing (if your car has this)
Check you have hot air coming from the heater - cold air usually means an air lock
Switch on aircon - does the fan run. Note that the fan will not run if the aircon system is not charged
Remove the connector from thermostat with engine running - fan should start

Hope this helps

White smoke would be a head gasket. In my own opinion. I have a coolant re fill kit that is a coolant system pressure tester that would tell me for certain.