Could be: shocks, tie rods, ball joints, steering damper.

Did it happen with the stock wheels?

Two thoughts:

1. These tires are larger than the originals and have a greater effect
on the car. So ANY problem will be magnified vs. an original size tire.

2. That the problem is temperature related points to the tires - tires
get flat spots from standing but such flat spots disappear as the tire
warms up from flexing while driving. These wide, low profile tires don't
flex much so it may take more than a km to warm them up. Performance has
its costs.

You can test this. Park overnight and, on a cool morning, before
driving, draw a chalk arrow on the tire to its ground contact point.
Jack up that wheel and rotate it so you can feel the flat spot at the
arrow (or not if there is none). Not scientific but may prove the point.

If it were my car I'd ask the Bridgestone factory representative to
check these tires and specifically comment on their characteristics vs.
#2 above. You may find out that "that all do that" - common auto
industry excuse to not deal with a problem - or, you may be given new tires!

Does this "wheels adjusted" means a full geometry Alignment, not just "in
line", but camber, caster, etc.? I guess so, judging by the amount of the
tires and rims purchase.
Did they inspect ball joints, bushes, tie rod ends, before the alignment?.
They usually do it, the car suspension must be in good shape with no worn
out parts, for proper alignment.

Though I don´t see a direct connection with them and the temperature
influencing the vibration, just in case, check the motor mounts. Do a search
in this group, there is a simple way to test them.

I just recently equipped my W210 and my W124 with Bridgestone Potenza tires
(japanese made). They have great ride, excellent grip and no flat spot at
all (it's winter down here, 2°C in the mornings). Anyway, see what happens
overnight with a higher inflation pressure, add 3 - 5 psi more than normal
(with cold tires).

MB: INSPECT DRIVE SHAFTS CONDITION. OUT OF BALANCE OR LITTLE
DAMAGE TO THIS AREAS CAN CAUSE SAME PROBLEM.(VIBRATION). GOOD LUCK

Thanks for your help.

I put up the tire pressure to 3,3. That helped so that the car didn't
shake when driving after a longer stop.

One wheel was sitting a little bit out of adjustment (1 mm at the tire
edge).
Removing a little bit of rust on the brake disk got it  down to app. 0,1 mm.

So now it seems to be OK.

I made a good tool for cheking. A wire wrapped around a stone with one end
sticking up like a swan neck. This I then put next to the wheel with the car
lifted
and I could easily see any misadjustment  of the wheel.

Michael