Think your diagnosis is correct, timing chain/tappets just pressurising on start up. Doesn't sound like anything to worry about as long as rattle goes quickly. These engines are sensitive to oil type and viscosity. I only use a good quality fully synthetic 5W/30 oil that meets the latest jag spec. I use GM 5W/30 (because I can get it cheap!) low ash oil, that exceeds all Jag, BMW, Merc specs. Made by Mobil I think. Don't be tempted to us Halfords Comma oils etc. Keep level at max on stick, dipped with oil hot after allowing no more than 3 minute drain down. I always used to dip the oil on a morning but was told this is wrong for modern engines using low viscosity oils as they hold a good half litre up in the top of the engine. Guy who told me this is jag/lotus trained engineer so think he'll be right. He also said I was wasting my money using engine flush or any additives as these high spec oils do the job even with 15k mile change intervals.
