The Smyth congregation (including Helwys) fled to Amsterdam to escape persecution. They worked and lived and worshiped in facilities provided by a Waterlander Anabaptist. Smyth obviously was deeply influenced by the Waterlanders, even so far as attempting to have his congregation joined the Waterlanders. Helwys disagreed with some points of Smyth's evolving theology, particularly unorthodox views about the celestial flesh of Christ and successionism in the validity of the ordinances and broke with Smyth.
Nonetheless, it appears that Smyth and Helwys were traditional Separatists (Calvinists) until they went to Holland, when they adopted an Arminian soteriology. It is not difficult task to think that they were influenced by the Waterlanders.
Helwys returned to England to found the first General Baptist church on English soil. The connection between the Anabaptists and the Generals seems fairly clear. That you can't find a document from Helwys that says "we owe some of our theology to the Anabaptists" is beside the point.
Now, is there any evidence that the Generals or Anabaptists influenced the Particulars? If you accept the Kiffin Manuscript and the Jessey Documents, it appears that the Particulars' rejection of infant baptism and subsequent requirement of baptism by immersion were strengthened by the examples of continental Anabaptists. The story goes that the English Baptists sent an emissary to Holland to learn about immersion and met with Collegiants, an amalgam of (largely) Remonstrants and Anabaptists — who were provided a declaration of faith by a Waterlander, Hans de Ries, who was eager for a union with the Smyth congregation.
Then I suggest your reading is not up to date. I am not arguing that the English Baptists were direct descendants of the continental Anabaptists; the Separatist/Calvinist leanings of the Particulars is beyond doubt.
Many years ago I described the second wave of the Protestant Reformation as a heady stew of diverse theologies and practices (some orthodox, some not) of which many groups — including the English Baptists — imbibed.
Nathan Finn, dean of the
School of Theology and Missions and professor of Christian thought and tradition at
Union University argues for a "convergent" view of Baptist origins, with influences from a variety of traditions merging with the Separatist river to create what we call Baptists.
Between The Times - Toward a Convergent View of Baptist Origins, Part 2