A member of the upper classes - usually male.
Members of the upper classes, by repute, often have minor genetic abnormalities like receding chins. This disparaging term is often used to describe members of the British upper classes and in particular the royal family. This is probably an implied reference to the effects of the supposed inbreeding of the upper classes and, again, particularly the House of Windsor. This is mostly just name calling, but is supported by the fact that Queen Elizabeth II and her husband have the same great great grandmother - Queen Victoria, and that she had a rather receding chin, as have several of her descendants.
Despite all that, this appears to be originally an American phrase. The first reference I can find in print is from a 'Mutt and Jeff' cartoon by Bud Fisher, in the New York newspaper The Post-standard, April 1910. This had the title line:
"Mutt starts barbering but doesn't stay - Strange how that chinless wonder hates poor little Jeff."
That doesn't give any indication of the meaning of the phrase, nor unfortunately does the cartoon itself. That meaning is strongly suggested in a piece from The Fort Wayne Journal-Gazette in September 1918 though:
"All you want is the Kaiser, ain't it? queried one soldier to a group of friends. How will the crown prince do? Maybe the chinless wonder will make a good souvenir."
Interestingly, this also refers to the lineage of Queen Victoria - the German Kaiser and Crown Prince were her grandson and great-grandson.