Wikipedia quotes founder member Martin Gore (guitars, keyboards, vocals and principal songwriter) as saying that the group's name "means hurried fashion or fashion dispatch."
He's only half right. As 80–89 (a website, cited by Wikipedia, that celebrates 1980s culture) points out, the French magazine's title, Dépêche Mode, simply means Fashion News. 'Fashion Dispatch' is admittedly a more direct translation, but to suggest that this implies speed, or haste, is wide of the mark.
80–89 also points out that "The confusion arises from someone remembering their high school French: the verb se depecher means to hurry". It suggests that the idea that dépêche mode is French for 'fast fashion' "is likely to have [arisen on] early (and now defunct) music sites." No doubt those sites did play their part; but if you ask me, so did the band's own spokesman.
© Haydn Thompson 2021