It was usual to translate the name of historical figures and monarchs until the early 20th century (as is still the case for popes) and a part of those translated names have become the usual vernacular names used to refer to such figures, both in French and in English.

For example, think about the tsars Peter the Great (Pierre le Grand), Catherine and Nicholas II (Nicolas II), known by their translated names instead of Pyotr, Yekaterina and Nikolai. You’ve probably heard of Catherine of Medici (not Caterina), John the Fearless (not Jean sans Peur), Casimir of Poland (not Kazimierz) or Frederick Barbarossa (not Friedrich) or Ferdinand and Isabella of Spain (not Fer(n/r)ando and Isabel). In the same vein, you can easily find 19th century English books about Lewis XVI of France.

Wilhelm, who lived during the period of transition when translated proper names went out of fashion for monarchs, was still quite frequently called "Emperor William" in English during his lifetime (for example) and "Kaiser Wilhelm" was popularised during the war as part of wartime propaganda aiming to make the German monarch feel more foreign.

Notice also that Wilhelm’s Allied contemporaries Nicholas II, Victor Emmanuel III (not Vittorio Emanuele), Peter I of Serbia (not Petar), and Constantine and Alexander of Greece (not Konstantínos and Aléxandros) are still known by their translated names in English, while Franz Joseph of Austria (a Central Power) isn’t.

Still, that French and English ended up cementing a different name at a time of shifting mores isn’t very surprising: both versions of the name were in use in both languages, and thus both were likely to be perenised by historiography. In this particular case (and that of François-Joseph d’Autriche-Hongrie), we ended up diverging.

This is an example of consonant strengthening or fortition. As a general rule, a word-initial /w/ of a Germanic word strengthens to /gw/ in Romance languages. Subsequently, it simplifies to /g/ in French. This explains partly how Wilhelm became Guillaume. Some other examples: werra > guerre, wisa > guise, want > gant.

Source: Ti Alkire and Carol Rosen, Romance Languages: A Historical Introduction, Cambridge University Press. See Section 3.4 in particular.