I posed this question today on History, and am searching for relevant phrases in English translations of 19th century accounts. Are there any other English phrases that I should be searching for, besides those with the root "immortal", that a translator might have chosen in place of "the Immortals"? I do not know how literal the translation from French to "the immortals" might be.
The French word immortel is probably the one behind the English "immortal" given that Wikipedia says it was the nickname of the Garde Impériale.
These words were both, on introduction, likely recherché, borrowed from classical Latin rather than having developed naturally in their respective languages (French etymology, English etymology). Often, such words enjoy great overlap in the modern languages that share them, and this is the case here too. Because of that, any English synonym for "immortal" is likely also a valid translation of immortel.
That said, the specific reason for the epithet, according to the French version of the article linked above, was « leur faible ratio de pertes au combat » : their low ratio of combat losses. Thus, if I were looking for possible alternative names, I would focus on shades of "immortal" that concern being untouchable — "invincible", "undefeated", etc. — rather than, say, being eternal.
Edit: To integrate jlliagre's comment into the answer, since 1833 « les immortels » has been used to refer to the members of the Académie française, derived from the motto « À l'immortalité » bestowed on its foundation, and nowadays the term mainly just refers to these members.