I have been following this thread with interest, but my time is limited due to my good wife struggling with severe post operative pain, the need to work on my own vehicles, and the demands of a large lawn. (The semi resident hare is nibbling away, but is severly over taxed.)
I have previosly suggested that codes P0215 (injection relay), and P0671 (fuel pressure regukator) could be due to a wiring problem. The U1600 code relating to the immobiliser system, and code P0481 (cooling fan relay) seem to be a new additions, and it also could be due to wiring problems., or more seriously there could be symptoms of water leaking into the ECU.
As has been suggested the U1600 errors could be due to low battery voltage affecting digital communication between the code receiver and the ECU. To elaborate, the code receiver interrogates the key, when ignition is switched on. It is the code receiver that controls the key code light. However again when ignition is switched on, the ECU sends a code to the code receiver, and if the key has been recognised, the code is repeated back to the ECU. If problems occur at this stage, it seems possible that the key code light could stay out, as the key has been recognised. I do not know that for certain.
The main injection relay operation is vital for engine operation, and located in the engine bay fusebox, where the fan relay is also housed. At ignition on, the ECU will expect to see 12V via both relay coils.
The fuel pressure regulator is also vital for the control system to function, as main variable quantities are fuel pressure, and injection duration. As stated in a previous post, eLearn for the 2.0jtd states that the ECU will shut down if the regulator cannot be seen by the ECU. It makes sense to extend this inhibition to the 2.8jtd, even though not explicitally mentioned in eLearn. On the 2.8jtd the hp pressure regulator is mounted on the hp pump, which in turn is part of the auxilliary assembly that carries the lubricating oil filter. The connectin to the ECU is direct, and not via the engine bay fusebox.
To summarise there is a collection of possible faults, with common factors being the ECU, and wiring close to it.
I have in recent months read of water getting into the ECU, but I think that was associated with snow blowing into the engine bay.
I also remember reading about damp cable forms adjacent to the ECU, suffering from internal wire corrosion, to the extent of going open circuit. (I thought PVC insulation was waterproof!)
I may be entirely wrong, but my current thinking would be to disconnect the battery, then ECU, and check for continuity and earthing on the relevant wires, perhaps starting with a loop test via the hp regulator coil, so no disconnection needed at the regulator. If the wiring to the hp regulator checks OK, then perhaps the ECU should be removed and tested, or the OP may choose to eliminate this possibility first.
I can post 2.8jtd engine wiring diagrams, and ECU pinout, but would need to know whether or not the engine has EGR option, and type of fuel filter - metal cannister, or UFI plastic bodied.