Source: lifelong bilingual in French & English.
As cced already said, common ways of expressing this in American English include “don’t worry about it”, “no worries”, “no big deal”, “no problem”, “you’re welcome”, “my pleasure”, etc.
French also has more built-in structures for formal address/politeness, though many of them are being left by the wayside just as we left feudal lordships with a peasant class.
For what it's worth, I'd say that the more "stuffy" version of "de rien" in French would be "il n'y a pas de quoi".
From nothing is an overly literal translation. You'll hear it in Spanish for sure as de nada (of/from nothing) or por nada (literally for nothing) depending on context though.