Keep at it your making progress! It takes some troubleshooting and focus on data and codes. All this noise stuff and phone videos is again useless.

Forget facebook and reddit. The land of morons. No your timing chain has nothing to do with this. Again stop listening to the wrong people. You have a stone cold expert helping you. Please listen to what Karl is saying! Please answer his questions.

Remember we are all just guessing on the internet! You need to help him help you. Step by step.
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You forgot to mention that I look good and smell nice. We will forgive your oversight this time.

 

Ooh, I liked your theory about clatter from the lifters, so I did a little more research. Seems both the V6 and V8 have solid lifters and are adjusted with shims. Nothing hydraulic to them, as found on most other vehicles.

But your line of thinking got me to thinking. Don't worry, it's just social thinking and I can stop anytime I want. Now my idea wouldn't explain everything, but maybe it could. Easy enough to rule out.

The timing chains have hydraulic tensioners to take up any slack. Guess what is used as the hydraulic motive force? Engine oil pressure. Purely conjecture here, but let's say oil pressure is lower at startup than typical, for reasons still undetermined. If this theory makes any sense, there could a little extra slack in the timing chain after a cold start. I doubt it would be enough to affect how the engine runs, but it could make a little noise.

There is another system that also uses oil pressure: Variable Valve Timing. Couldn't find a lot of details about what the default position would be if oil pressure was below normal. Could this affect starting? This is just me thinking out loud, trying to connect the little puzzle pieces. Mostly just wondering if it could all be tied together, both the noise and rough running after a cold start.

But what to do about my latest crackpot theory? OP did mention having changed the oil and filter. My very first thought there was the filter did not have an anti-drainback valve, or if so equipped, it was sticking open. If the oil filter drained back overnight, the engine would have to run for a little bit to refill the filter and then send the oil out to the various destinations. So my latest idea, subject to medication levels and wind direction, would be to replace the oil filter. Do NOT use the same brand presently installed. Be triple sure the new one is listed as having an anti-drainback valve. Maybe a $15 gamble to see if a new filter helps?

Whaddya think?

Remember, we're all just guessing here. I'd certainly try a $15 oil filter before a $2000 fuel pump.

 

My other thought, of a leaking fuel injector. If an injector didn't fully close, it could dump fuel into a cylinder. This floods the cylinder walls and washes away the oil, decreasing compression. The spark plug may also get fouled and not fire properly.

A basic test is easy enough. Pick a time when you won't need to drive the car for several hours. Set up your scanner to read fuel pressure. With the engine cold, start the engine and watch the fuel pressure. For troubleshooting, it would actually be helpful if the engine runs roughly at first. Watch the fuel pressure and see if it fluctuates. This isn't even the actual test yet. Just establishing a baseline.

Let the engine warm up enough so it stabilizes. Continue keeping an eye on the fuel pressure. Shut off the engine. Quickly turn the key back to Run but not Start. You want the computer powered up so the scanner can read fuel pressure. The value should be stable, with maybe just a slight decrease. I don't have any specs on how much of a drop is too much over X amount of time, so we'll have to use our judgement. Turn off the key now.

Next, open the trunk and lift up the trunk floor. In the rear power distribution box, remove fuse F47. This is the power source for the fuel pump and we don't want it to run for the test. After a couple of hours, turn the key back to Run, but still not Start. We're only doing this so the scanner can read fuel pressure. Normally the pump would operate for a couple of seconds with the key to Run, to pressurize the injector rail before starting. But we pulled the fuse to prevent that. We're only interested in seeing how much the pressure decayed since the last reading.

Won't cost a penny to try this test, just a little time and patience. If the pressure drop seems excessive, we can figure out some means to isolate where the pressure is escaping. Could be an injector leaking. Could be a bad check valve between the pump and injector rail.

 

@ Karl: Yes, sorry about the hydraulic lifters - did not know, we don't have them in our Jags - I have them in my Ford Fairlanes, and I even swapped them already.
But yes, I agree with the hydraulic tensioners, etc.
Also a good theory about the oil filter - the question. if an oil filter had been used that DOES HAVE the anti-drain-back valve. Thus, which filter did you use, Colin? The internet should give the answer, if that specific filter does have that valve. I have just spend recently several days to find the correct filter for my 3.0L 2004 S-Type (which btw is the same for X-Type), and I certainly came across the issue of the anti-drain-back valve.

@Collin: Yes, obviously you can't be starting the car and at the same time as listing for noises in the engine bay - I thought it to be obvious that someone else needs to start the car, while you are listing up front....

 

Another option is to have himself cloned. Then he'd have a helper available at all times.