I assume the nonpolar substance would move just as much in methanol as in diethyl ether. The diethyl ether isn't making the nonpolar move any faster, the nonpolar having no attraction to the stationary phase is why the nonpolar has a high Rf value. There's basically 2 ways of looking at it and both end up with the same qualitative result.
Way 1: In the nonpolar solvent, the nonpolar substance is binding to the nonpolar solvent, making it move more and giving it a higher Rf value. Probably true.
Way 2: In the polar solvent, the nonpolar substance is not bound very well to the mobile phase, however, there are also going to be way less available binding sites on the stationary phase because now the polar substance AND the polar solvent are going to be trying to bind to the stationary. So, you might think that the absence of the nonpolar solvent is going to hurt the nonpolar Rf value, but in reality it should probably stay about the same.
And yea, you don't add solvent to TLC, you would only add different solvents to column chromatography to get things to continue to elute.