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There are 3 main Biblical Unitarian views of John's prologue (John 1:1-18).
The first is that the logos = God's plan or thoughts, and the beginning is the old beginning (Genesis 1).
The second is that the Logos = a personification of God's word, and the beginning is the old beginning (Genesis 1).
The third is that the Logos = Jesus (similar to how Jesus is the Light, the Bread, the Gate, and so on throughout John - it is a title or identifier which refers to Jesus), and the beginning is the new beginning (Mark 1:1, 1 John 1:1).
In all these cases, Biblical Unitarians (BUs) do not hold that John 1:1-13 refers to notional pre-existence of the logos. In the first two, God's plan or word really did pre-exist Jesus. In the third one, the beginning is Jesus' ministry, and so doesn't pre-exist Jesus.
So, for BUs who hold to one of the first two, the logos ('Word') of John 1 is not identified with Jesus. Rather, the understanding of 'Logos' is along the lines of an idea, plan, or indeed speech (as in Genesis 1, where God speaks things into existence - it is God's creative power). So at John 1:14, "the Logos became flesh" is understood similar to "the plan took shape" or "the idea became the cup" or even "the score became music" or "the script became a play". That is, the logos leads to, informs, and is reflected in Jesus. In this sort of sense, the Logos became flesh.
On the first two views, then, Jesus (= the Christ) comes to exist at his conception, which is held to be John 1:14, and they would hold this is then describing in different language what is also described at Luke 1:35. The lead up to this is a bit complex interpretively, because of the vagueness of John's language, but throughout John 1 these BUs typically hold John is describing what actually happened (the logos actually was with God, things actually were made through the logos, John the Baptist actually arrives on the scene, and so on).
This talk by Dale Tuggy (a BU) is a good discussion of John 1 from this sort of BU perspective, including an historical survey of interpretations, including a useful handout for the talk.
On the third view, however, John 1:14's kai ho logos sarx egeneto refers not to Jesus' conception (which is never discussed in John's Gospel) but rather to the beginning of Jesus' ministry (and so dovetails with John 1's 'beginning', which on this view is also the beginning of Jesus' ministry).
Also see 1 John 4:2, "Jesus Christ has come in the flesh [sarki]," similar to John 1:14 and to Weymouth's translation of John 1:14 ("And the Word came in the flesh"). See Jesus' own "for this reason I was born and came into the world" at John 18:37 - the coming appears to happen after his birth, not before it, and is consistent with the start of his ministry.
So naturally, Jesus begins his ministry and then 'tabernacled among us' (also in John 1:14, immediately after 'kai ho logos sarx egeneto') as an itinerant teacher (similar to how the Angel of the Lord tabernacles among the ancient Israelites as they wandered about).