If the engine spins fast, that pretty much eliminates the oil. (Slow cranking can inhibit cold starts, but it's rare.
White or gray smoke is an indicator that perhaps the glow system (intake heater if it's a DI diesel or precombustion chamber glow plug if it's an IDI) is not working well, or that there is air in the system. If the probelm doesn't occur at all in wrm weather, that points to the glow/ intake heater system and you should have that rechecked. If it happens at any temperature, that leaves air as the more likely possibity.
Small leaks on the suction side of the fuel system can introduce air into the system. If the car is harder to start the longer it sits, that's a good clue. In your case, when the technician saw the car, it had been started and had been bled of air... no symptoms for him to trace back. We wpould assume he properly checked the glow/intake heater system.
I would test this myself. Run the car and then let it sit in the cold until it has completely cooled off ( 5-6 hours in cold weather at least). You might drive it in the evening and start it in the morning. If it starts well after a few hours of sitting but not after days of sitting, that might be a good indication of air. Try that a few times and if no trouble, let it sit longer until you experience the trouble. If you get trouble right away, you may have a big air leak but it also point bclk to the glow/intake heater system.
I don't know your car at all but the way I test for air in diesels is to install a loop clear plastic line in the low pressure part of the fuel system, just before it enters the injection pump, bleed all the air, then observe it for significant air bubbles. If your diesel is a common rail, electronic system with no injection pump, this will not likely apply.
No matter what, if it's spinning over fast, I doubt it's the oil but it still never hurts to get better cold weather oil flow.