With Christianity legal, there was also the issue of what to do with the barbarians who invaded the empire. People evangelized to them either because the Church was obeying Jesus, or because the government thought it would bring peace to the empire. This pretty much began around the time of the Edict of Milan.
Goths first showed up in the Danube region. They asked to enter because they were being chased by Mongols. This caused thee battle of Adrianople where emperor Valens died and Arian Visigoths came to live. They sacked Rome and then founded a kingdom in Spain.
Later, Arian Ostrogoths took over the leading of the Romans. Also, the Burgundians and Franks moved into Gaul. All these followed Arian Christianity which is not Christianity. It's the belief of Arius that Jesus was not equal to God the Father, but subordinate to him. Hopefully, Cairns will address the Council of Nicaea in later chapters, but for now a bunch of non-trinitarians populated the empire.
Gregory the Illuminator successfully evangelized Armenia. Frumentius established the Coptic Church in Egypt and Ethiopia. Cairns is not quite sure how the Gospel came to the British Isles, but they came to know Christianity. Sadly, these folks did not submit to the Pope, and they gave us heretics such as Pelagius who believed we had to find our own path to God based on Jesus's example. They were easy bait for the pagan Angles, etc., who drove them from the country.
Ulfilas witnessed to the Visigoths around the Danube, but sadly, he was an Arian, so they also had the wrong idea of Jesus. Many Romans took on the task of converting the Visigoths and Lombards from this counterfeit faith to the true Christianity.
When the Teutons invaded the Rhine area, Martin of Tours, the patron saint of France, converted many of those people, and at some point Clovis, the emperor, became converted to the Faith. St. Patrick was kidnapped by pirates and taken to Ireland, and then he returned and evangelized those people. Columba also did the same thing to Scotland, founding a monastery. Soon, the Nordic Europeans grew to know of Christianity. Some people, however, brought their hero worship into the faith and converted it to saint veneration. This would have a poisonous effect throughout history.