...

In colder weather mine usually clatters for a few seconds on startup.
Sometimes it lasts a few minutes until I get out on the main road and speed
up.  It doesn't bother me at all and I don't worry about it.  It has 164k
miles and has done it from at least 100k.

The thinner the oil, the less it will happen.  I usually use 15W40 because
it's cheaper and I feel more comfortable with thicker oil.  With 10W40 the
problem is less.  I had some Castrol RS 0W40 from Halfords for half price
and the tappets never made a sound.  However, 1. it's £35 a bottle at full
price, 2. the bottles are only 4L so you need two for each oil change, 3. my
engine does use some oil, although it doesn't smoke, but with that stuff I
got through nearly four bottles before the oil needed changing again.
Shame, because I genuinely think the engine had a bit more power.

I've never had the noise come on after the engine was started.  With yours
being a vanos version it could be that and not a tappet.