I would say smoke smell is also a bit like diesel smell

I will ask garage to do vacuum hose check

Would decrease or leak in vacuum pressure cause any issues

I will re check colour of oil to confirm

I was told it was Castrol oil

Thanks every so much for your help so far

Just checked oil again it's not black more browny grey is

Initially the first lot on dipstick was very dark like a black colour

Vacuum pressure is very important Any leakage would cause performance and emmissions issues as turbo is vacuum controlled and the so is the EGR valve,by solenoids.(electric switches).
Even your engine mounts are vacuum controlled!When the mechanic does the vacuum test ,he will see vaccum hoses going to left and right engine mounts(these are only on auto transmissions X5s!)

Ok..dark brown colour on oil is good,just ask them what viscosity oil grade they filled up?Both 5W30 or 0W40 are fine.

If smoke smell is also diesel,just confirm the glow plugs again.You may also have a leaking injector but this should have been diagnosed!A leaking injector leaks mostly when engine is stopped and fuel is deposited in chamber,so when you turn engine on,this will come out of exhaust as unburnt fuel.!Glow plugs as you know,are there to heat up fuel in sub-zero temperatures ,so this time of year they are only on standby,really.

Get the mechanic to remove the turbo outlet pipe ,(the black rubber 90 degree pipe tha is directly under the air filter box) and check for excessive oil in the turbo and the play in the compressor!

Thanks I will get them to check this all out during this week and report back

Your signature doesn't mention what year or engine on your E53. You have a diesel (so do I). When you start the car, do you first wait until the DDE (yellow) light goes out before engaging the starter motor? Does outside ambient temperature make a difference in the amount of smoke you see? Do you have this when you restart the car when the engine is warm?

If you replaced the "loo roll" breather with the vortex, that's a good long time upgrade (I've done the same). How clean is your air filter? What specific error codes do you know have when you read codes?

Oil, Castrol brand and weight isn't enough to be sure the mechanic put the correct oil in your car. It MUST be type that is BMW LL04 specification approved. Only a specific number of Castrol products meet this.

Yes wait for light to go out

It smokes less when start engine again if it's warm

Car 2004 e53 diesel m57n

Car was serviced two days before purchase and I was told its bmw spec

+1 on air filter - verify that it's clean.

It had new air filter, cabin filter, fuel filter and oil change 1000 miles ago

I changed oil breather to vortex also

I don't have any experience with the m57n but they appear to have a common issue.
BMW EGR bypass kits : BMW EGR bypass M57N 330d 530d 730d X3 X5

What common issue is that?Thats an aftermarket egr bypass kit!