guys, why no one here mentions oil grades just goes to show we have a long ways to go in bringing up our knowledge and understanding of engines.
first, OP: what is your oil type used.
second, many people have reported online discussions switching to 15w-50 eliminated the rattle on start up. they have multiple reports of replacing tensioners, guides, etc with no significant changes.
third: oil filter canister. people say that the inside piece breaks down and allows oil to drainback to sump which doesn't give the tensioner oil flow at startup.

fourth: lack of information. I have seen an unlimited supply of losers on other forums and sometimes here, who like to show up to hit and run their own threads too, so there's that.

I do see a TSB for the E39 for rattle at start but that only applies to vanos rattle on the S62 engine, which uses different vanos system than the M62. SI B 11 04 01 .

Attached is the breakdown of the tensioner. I found that it's challenging to install once removed. You can crank the engine clockwise a 1/8 turn to release some tension on the chain of the right bank of the engine, and using a socket press into the threads and turn until engages. We can also check that the spring is facing the right direction, and install a new oring or crush washer. I use what's called 'bonded oring' which I just got from hydraulic supply industrial supplier. I believe it's metric size 24. You shouldn't need to replace this part unless there's some scoring on the cylinder or bore, or if the spring has cracked.

Attached Images

 

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2003 4.6, esotril blue, over 215,000KM. Every previous owner failed to keep up the vehicle maintenance. Restoration project.