This question comes up in The People's Guide to J.R.R. Tolkien. Stating briefly only the Witch King and two others went into the River. The river was commanded by Elrond. Remember Elrond has the Ring of Power, the Water Ring. The Nazgul could probably sense this power and only their connection to The One Ring would have drawn them into the river.
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By the starlit mere of Cuivienen, they rose from the sleep of Iluvatar
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