The dvadashakshara is not Vedic; its earliest secure appearance is in the Bhagavata Purana, a text scholars generally date to somewhere between the ninth and tenth centuries CE, composed in South India and drawing together far older Krishna material into the form now read across the subcontinent. Its transmission afterward runs through exactly the channels bhakti literature describes for itself: guru to disciple, bead by bead on a mala counted to a fixed daily number, a practice carried forward unbroken into Gaudiya Vaishnavism and, in the twentieth century, into the international Krishna consciousness movement, which uses this mantra alongside the longer Hare Krishna formula rather than instead of it.
The translator's problem sits in one word, Vasudeva. Read as a patronymic, it simply names Krishna as the son of Vasudeva, his human father in the Mathura story, so the mantra becomes homage to the god by his family name, an oddly domestic frame for cosmic address. Read the other way, splitting vasu and deva, it becomes he in whom all things, all vasu, dwell, an etymology the Puranic and Vedantic commentators themselves offer and one that turns the same word into a metaphysical claim rather than a genealogical one. The Bhagavata Purana appears to intend the pun deliberately, letting the ordinary patronymic carry the philosophical reading without ever having to choose, which is characteristic of how Puranic Sanskrit works and is largely lost whenever the mantra is rendered into a single fixed English gloss.