What imperial machine screw has a diameter of 4.5mm and a thread pitch of ~1mm?
A friend of mine pointed out that #9-24 UNC is actually a standard size – it is far from common these days (mind this is an old piece of furniture). Here is a table from 1916:

The listed diameter of 0.176 inch amounts to 4.48 mm – well within my range of measurement error.

Now that this is cleared up, the real struggle will be to find one of these ancient screws, with sufficient length, more than 100 years later… in Germany.

On a side note: I did read the other answers and I appreciate them. I got my hands on a #10-24 UNC screw, but it does not fit.