2005 Land Rover LR3 diesel, turbo take a long time to pickup speed and there is a lack of power and lot of black smoke,what is the cause

I am assuming you have a diesel as based on your description, the most likely cause is a split or crack in a turbo hose. You will also get black smoke if that is the cause.

Get this to a Land Rover tech asap.
Module could be a central convenience module or body control module. Either way, wiring in these vehicles can be very tricky. There are just certain vehicles you don't mess around with unless you have all kinds of time on your hands. Bring it to a certified Land Rover garage. Could definitely be the basis for all of your odd issues.

Distributor? Dist. rotor? Wires / Leads? Plugs? ....

it is time for an alternator,,,real common in the lr3.

The issue is the engine is running rich (too much fuel or not enough air) So the things that you can check are the air filter, oil level, coolant. The rest is determined by diagnostics to see if the turbo is not producing enough compression or the injectors are worn

some of these cars have a plug near the trip meter reset that is pressed with a long narrow object like a thin Philips screwdriver or a knitting needle.

sounds like a new turbo job

Service light reset – Land Rover PDF File

I hope helps with this (remember rated this help).

Hi ,
Not easy this one,when your turbo failed the oil feed to the turbo bearings continued to feed engine oil into the intake side(could not turn off),the engine is then Un-governed and free to rev way above maximum limit,it has then massively over heated, the oil treatments are unlikely to cause this problem ,usually down to not letting the Turbo "wind down " after driving,The best course of action is to replace the engine and turbo charger and have the induction pipes and intercooler properly cleaned ,there are no cheap fixes for this you were just unlucky.

Check the breather filter, a quick test is pull the oil dipstick and rev engine if oil rises up dipstick replace breather filter straight away as it will blow engine if not