Its arrival in Europe and America can be dated to a single voyage.
A. C. Bhaktivedanta Swami Prabhupada arrived in New York in 1965, at the age of sixty-nine, on a cargo ship, with a trunk of books and almost no money. He chanted in Tompkins Square Park, attracted a following from the countercultural scene of the Lower East Side, and founded the International Society for Krishna Consciousness in 1966. The movement grew very fast in the following decade and the formula became, briefly, one of the most recognisable religious sounds in Western popular culture, recorded by George Harrison in 1969, sung in a Broadway musical, and audible in airports.
Two things are worth noting about this. The first is that Prabhupada was transmitting a specific lineage, the Gaudiya Vaishnava tradition through the Gaudiya Math, and not Hinduism in general; the theology is a particular one, and the identification of the movement with Hinduism as such, common in Western reporting, misrepresents both.
The second is that the movement's subsequent history includes serious institutional failure. Investigations and litigation from the 1990s onward established patterns of abuse in its boarding schools in India and the United States, and the organisation has acknowledged this and paid settlements. This is a matter of court record rather than of allegation, and an account of the formula's spread that omits it would be incomplete.