I have exactly the rattle outlined in the OP, and at the same times, under the same conditions.
I bought my car at 40K miles, on a warm early October day. No hint of noise on start-up or while running. A few weeks later when the weather turned colder, I started noticing the rattle. I wondered whether it was a heatshield issue (had exactly the same type of noise with my old TS 156). But apparently not.
The noise completely disappears when the engine is fully warmed up, and doesn't really show itself at all during warm summer months. When the weather turns colder, it's there again, on start-up. I've mentioned it to both (very good) independent garages I use for servicing but of course by the time I've got to them the car's warmed up!
The car runs just fine, and I've now completed 30K miles with the car (now totalling 70K miles). It's never got any worse. AFAIK the previous owner was just as fastidious about servicing as I. But he'd only had it from around 25K IIRC.
Well. Because it's easy and cheap I decided I'd chuck a £5 bottle of Wynne's Hydraulic Lifter Treatment into the oil. I have to say I have zero faith in oil additives. Years ago I worked regularly with Bike magazine and we had a whole bunch of them lab tested, and none came out with a clean bill of health, varying between snake oil and sulphuric acid. Whatever tiny additional protection they gave was usually at the cost of much greater corrosion etc, and especially no good for bearing shells. But this Wynne's stuff has quite a lot of positive feedback and is supposedly just detergents - a less aggressive form of flush.
Within a mile the tapping was much quieter! By the time I got home (4mls) I had to listen quite hard to identify it at all. Not only that, the entire top end is significantly quieter and smoother sounding, and I hadn't even realised it was rattly. The single noisy tappet is
still there, but it no longer sounds alarmingly louder than the rest, just a little distinct from the healthy background rattle. The sharpness of the noise has reduced. It's much less and sounds dampened, presumably by better oil flow.
I am not calling it fixed, but this does rather confirm what I thought: gum and sludge was preventing the tappet from working properly.
So far (only 4mls) that Wynnes stuff gets about 7/10 from me.
I'll run it like this for a while to see whether there's any further improvement, then use a flushing additive and change the oil and filter. Possibly it having been run like this for a few thousand miles will have worn at least the tappet and it will need to be changed, but I'm not going to be doing a lot of miles over the winter. If it stays like this or better, that can all wait for warmer weather.
Now it's a bit quieter, I can hear other stuff to be paranoid about as I get to know the car. I'll post some more questions in new threads.