This is a great question, thanks for asking it.

Let's start with some background. Nestorius was an Antiochene monk who had been installed as the Bishop of Constantinople in the Spring of 428. Early in his career, he preached against the title θεοτόκος (Theotokos or "God-bearer") for Mary, the mother of Jesus, offering instead the title of Χριστοτόκος (Christotokos or "Christ-bearer"). The latter, Nestorius thought, preserved divinity from being implicated in birth and death (and thus change). Divine impassibility was one of Nestorius' principal concerns, obviously. But it should be noted that Theotokos was a pious title in use since at least the 3rd century (we find it in Origen as well as the hymn Sub Tuum Praesidium).

According to Cyril of Alexandria's letter to Pope Celestine (Ep. 11), Nestorius allowed Bishop Dorotheus from Marcianopolis to come in and preach in the summer of 430. In his preaching, while Nestorius was in attendance, Dorotheus anathematized anyone who would call Mary the mother of God. Socrates Scholasticus relays a similar story, saying that Anastasius, a presbyter in Constantinople and Nestorius' right-hand man, had preached against Theotokos arguing that "it is impossible that God be born of a woman." (Socrates Scholasticus, HE 7.32).

The Council of Ephesus was called by Theodosius II in the winter of 430. Nestorius had already been condemned by Pope Celestine in the West at a Roman Synod held in August of 430. At the council in 431, Nestorius was deposed and Cyril's concept of the hypostatic union won out, understood as the better interpretation of the Council of Nicaea (the chief authority in late ancient Christianity next to Scripture).

This, however, did not settle the matter. Far from, in fact. As Jaroslav Pelikan writes:

That was anything but the end of the christological controversy. As soon became clear, the Ephesian resolution of the conflict was not acceptable to anyone and had itself to be resolved in one or another direction.

Thus, in the intervening twenty years, it became clear that Ephesus had satisfied neither the Nestorians nor Cyril's party. A new council was called in 451, this time at Chalcedon, to settle the matter. Nestorius had been in exile in the Great Oasis in Egypt between Ephesus and Chalcedon (spending a few years initially back at his home monastery, but moved to Egypt in the early 430s).

This question is particularly interesting because, in fact, the West never really saw Nestorianism take off. The West had, since the time of Tertullian (d. 220AD), held to a two-substance, one-person Christology (cf. his Adv. Prax.). Because the West had a stable concept of person (persona), it was never susceptible to the central issue in Nestorianism, namely assigning predicates ultimately to natures instead of persons. They always had a single person who was the subject of attribution. Thus, in the West, it was common to say that "God was born" (Deus natus est) or "God suffered" (Deus passus). The Westerners did not, of course, mean that divinity suffered - that would simply be to mirror Nestorius. Rather, they could say that "God suffered" because Jesus Christ, who is one person, is both God and man. He suffered, and since he is God, it is right to say that God suffered. This is what is known as the communication of idioms (communicatio idiomatum).

The only trace of "Nestorianism" in the West isn't really Nestorianism, but precedes Nestorius by about 10 years. It's found in the Libellus Emenedationis of Leporius, cited by John Cassian in his treatise against Nestorius, titled, On the Incarnation of the Lord Against Nestorius in Seven Books. In Leporius' Libellus, there is a key phrase wherein he says that:

quasi inferiora se deus agere videatur, ita hominem cum deo natum esse dicamus, ut seorsum quae dei sunt soli deo demus, et seorsum quae sunt hominis soli homini deputemus

Lest God be thought to do things beneath him, we said that a man was born with God, so that those things which belong to God alone may be said of him separately, while those things which belong to men alone could be said separately of the man

This is essentially Nestorius' dyophysite hermeneutic at work, parsing the predications according to natures rather than persons. But this is as close as the West ever gets to "Nestorianism", at least before Nestorianism. It never gains any ground in the West simply because Latin theology had this concept of "person" that the East did not figure out until the fifth century.

For further reading:

Bethune-Baker, J.F. Nestorius and His Teaching: A Fresh Examination of the Evidence (Eugene, OR: Wipf and Stock, 1998) [This book is mostly bad, but it ought to be read if you're interested in Nestorianism]

Loofs, Friedrich, Nestorius and His Place in the History of Christian Doctrine (Cambridge: Cambridge University Press, 1914). [Loofs is slightly better, but still too sympathetic of a reader to Nestorius such that he is often blinded to the deficiencies in Nestorius' thought]

Pelikan, Jaroslav, A History of the Development of Doctrine, Vol. I: The Emergence of the Catholic Tradition (100-600) (Chicago: The University of Chicago Press, 1971).