330d sport smoking and overheating.......help

When the head gasket fails on a liquid (water) cooled engine the typical smoke colour is white (steam) and that becomes rather underlined by the steady drop of the coolant level and often clear evidence combustion gas is leaking into the cooling system.

Smelly black/grey smoke tends to be produced by an over-fuelling condition such as a stuck choke or strangler or other cold starting aid or perhaps carburettor flooding.

Blue or blue/grey smoke is invariably engine oil burning but I suggest you ignore that until the black smoke problem is rectified and after a short time the blue smoke might also disappear. If it doesn't it could be related to the overheating but the black smoke probably wasn't caused by that...

blown head gasket

Your engine would smoke if you are running at too high RPM or have had your RPM at a high level before. The engine is getting overheated. When you say it only smokes until about one mile it sounds like a problem with the engine. starter.
Source here ...

overhet then smoke as I'm sure you know is not good. then shut off and will not start, sounds like a blown head gasket, this would allow coolant to escape to the point of overheating, and the smoking as well, coolant in the cylinders via a blown gasket would prevent starting. DW Mylin

TIme for an overhaul and rebuild or a new engine. The rings and seals are shot. This is not going to be cheap. IF you keep driving it, there is an outstanding chance that you will seize the engine

Valve stem oil seal is worn. It doesn't smoke when injector disconnected because no compression as cylinder not firing with no fuel.

Sound like you have no boost. Turbo or actuator faulty

What do you mean remapped?

If you mean retuned, then the new Convertor and Flywheel must have thrown the adaptive learning off. timing and Ignition is off now, you are not burning all the fuel that is being Squirted in the Motor. Have the tuner return it to OEM and then retune it. bc to me it sounds like that is the Problem. Did you put the same donut back in and the same oem flywheel? or did you go with a stage kit? Bc these beemers are very sensitive. Have that checked out. Let me know waht it was.

The most likely senario is that the water pump went bad which caused the overheating in the first place. Then after the engine shut down, the water pump seized up. When the engine was restarted, the timing belt started to smoke because the water pump, which runs off the timing belt, was not turning. Since the engine overheated to the point of shutting down, blown head gaskets are a possibility. Put a new timing belt and water pump on it and start it up. If it runs OK, you're good to go.

the white smoke is oil smoke,check your valve stem seals,if they are bad it will smoke till it burns the oil that leaked by when it was sitting.as far as it stalling in the morning check your choke it may need adjusting.