Thanks for the comprehensive reply, Andrew.
The reason I asked is that my son's 2001 406 has a similar problem, that I was trying to figure out a couple of years ago. But he lives elsewhere now, so I've sort of forgotten about it.
When the engine is started from cold, the radiator fans run immediately, the temperature gauge needle stays in the OFF position, off-scale on the left, and the dashboard screen has the warning, Coolant level low, or something like that.
Then, after about 5 or 10 minutes of driving, and the engine has warmed up, the needle suddenly flies up to the hot end of the scale, then settles back to normal, and the low coolant warning disappears.
Cheers,
Warwick