If the projected points could be returned to latitude and longitude then anything can be done with the latitude and longitude. How is the map projection unknown ? Where is the input file ?
However, the Robinson projection has more of a 3D effect than a flat map view. It looks similar to a simple "cosine" projection for half-the-world but with the top cut off:
Convert spherical latitude and longitude to rectangular coordinates in meters as: North coordinate = Latitude * 1852 * 60 and East coordinate = Longitude * 1852 * 60 * Cos(Latitude) . But first translate all the longitudes by the same constant so as to straddle zero longitude. Also, adjust for radius by changing the number of meters-per-minute of latitude.
Except to make the 3D effect don't translate to a zero longitude zone, don't make zones, but just run the numbers.
Then the Robinson projection has less severe convergence of the wide-side longitudes at full-width-world because it's not aiming to converge to the pole.
Basically, for a whole-world 3-D effect view then look up the Robinson formulas.
Or make a cookie-cutter map of the whole world with UTM. Set the UTM zone centers parallel from the equator and then plot the coordinates of each zone from each zone center. That makes a map with cookie-cutter types of spaces but accurate if not scaling across the spaces:
But ignore specially sized zones as that is MGRS and not UTM.