For those who deny the Immaculate Conception, why did Jesus not have original sin? The Westminster Confession (8:2) says Jesus is of the substance of Mary:


  The Son of God, the second person of the Trinity, being very and eternal God, of one substance and equal with the Father, did, when the fullness of time was come, take upon Him man's nature, with all the essential properties, and common infirmities thereof, yet without sin; being conceived by the power of the Holy Ghost, in the womb of the virgin Mary, of her substance. So that two whole, perfect, and distinct natures, the Godhead and the manhood, were inseparably joined together in one person, without conversion, composition, or confusion.


Calvin gives his interpretation of 1 Corinthians 15:22 in Institutes book 2 chapter 1 section 6:


  Therefore, the only explanation which can be given of the expression, “in Adam all died,” is, that he by sinning not only brought disaster and ruin upon himself, but also plunged our nature into like destruction; and that not only in one fault, in a matter not pertaining to us, but by the corruption into which he himself fell, he infected his whole seed.


If Jesus is of the substance of Mary and Mary (a seed of Adam) is infected by Adam. How can Jesus have no sin?
There are at least several ways that this is explained.  Here are three common ones.

One approach, taken by early and medieval theologians before the doctrine of the immaculate conception was widely held, is described by Thomas Aquinas:


  As Augustine says (Gen. ad lit. x, 20), Christ was in Adam and the other fathers not altogether as we were. For we were in Adam as regards both seminal virtue and bodily substance, since, as he goes on to say: "As in the seed there is a visible bulk and an invisible virtue, both have come from Adam. Now Christ took the visible substance of His flesh from the Virgin's flesh; but the virtue of His conception did not spring from the seed of man, but far otherwise—from on high." Hence He was not in Adam according to seminal virtue, but only according to bodily substance. And therefore Christ did not receive human nature from Adam actively, but only materially—and from the Holy Ghost actively; even as Adam received his body materially from the slime of the earth—actively from God. And thus Christ did not sin in Adam, in whom He was only as regards His matter. (Summa Theologica, 3rd part, Q15, Art. 1, reply to objection 2)


So to Augustine and Aquinas, the invisible, spiritual side of Christ's human nature came directly from God, not Adam.

This is somewhat similar to the view of the Anabaptists, which believe, as summarized by Louis Berkhof:


  The prevailing opinion among the Anabaptists was that the Lord brought His human nature from heaven, and that Mary was merely the conduit or channel through which it passed. (Systematic Theology, 3.2.1.B)


But in the Reformed tradition, greater emphasis is placed on Christ's human nature coming from his mother, and as a result his sinlessness is instead attributed to the sanctifying work of the Holy Spirit.  He thus has two roles in the conception of Jesus – first, as the "efficient cause" of the conception, and second:


  He sanctified the human nature of Christ in its very inception, and thus kept it free from the pollution of sin. (Systematic Theology, 3.2.1.B)


So we see that there isn't a single answer to the question, but rather a variety of ways to explain how Jesus could have human nature and yet be sinless.