3.6 TDV8 smoke and rough idle help

I’ve just bought a 2010 L322 3.6 TDV8
It was more of a heart over head purchase. when I turned up to buy everything was great apart from the fact that it had been warmed up and was idling when I turned up.
at the time I thought that he was trying to hide something but the car was in great condition with full dealer history and in the exact spec and colour combo I wanted and a good price.

the problem:
when you start it from cold it has a rough idle and there’s a white smoke from the exhausts for about a mile or so down the road it continues to smoke.
It is also very hesitant if you try to put your foot down.
Then once warm the car idles beautifully and doesn’t smoke at all even if you floor it.

There are no splits in any turbo hoses.
It doesn’t have any fault codes and doesn’t ever go in to limp mode.
If I take the two turbo hoses off the top of the intercooler the turbo on the left side is blowing through a slight misting/smoke but not much with a tiny bit of oil if you wipe your finger around the inside of the hose.
The other side doesn’t have any misting or smoke and feels like it’s blowing stronger.

However with the pipes disconnected it still smokes at idle.
It didn’t use any coolant or oil on the 500 mile trip home from buying it.

Thanks in advance

The white smoke may just be condensation in the exhaust, my diesels do that.
No idea about the rough idle, but if it came from a dealer, take it back, it's up to them to fix it.

Could be egr valves playing up ,think there prone to it

Same symptons was injectors on my d3.
Also known to cause rough idle on the d3 (though 3.6 may be different) is the oil temp sensor on the side of the sump, plug in code reader and see what it reckons the oil temp is.

If the smoke smells like a strong diesel odeur the injectors may need testing. It seems to me they warmed that engine up for a reason and your hunch was correct. If you've not had it long, take it a garage for a quote and then approach the seller with the result from the garage and start asking questions as to why it was warmed up before you got there?

Back to the dealer, definitely.

Could be the injector is leaking when parked up so it has to burn iff the excess fuel after starting.