According to my client, who has a copy of the original atlas, a bit of text in the index refers to the Winkel Tripel projection. I've created a graticule that matches the one on the map, limiting it to the "whole" cells (so from -120,-80 to 160,80) and overlayed that in MAPublisher, this is what it looks like.
So it's pretty spot on along the equator but gets progressively worse towards the poles.
Played around with the latitude of origin and after some trial and error I arrived at 42, which seemed quite appropriate
Not quite there yet as you can see, but considering I have no idea when, how and to which standards the original map was produced I'm calling this "close enough" for now.
Thanks for your suggestions Daan and Souvik!