As early as 60 AD, Paul the Apostle indicated an interest in visiting Spain. [The book of Romans 15:24.] At this time in the Roman world, the "Diocese of Spain" was administratively considered part of the "Prefecture of Gaul". In this Prefecture, besides the Diocese of Spain, and the Diocese of Gaul, was the "Diocese of Britain". It was the increase in persecution of this new "sect" that would lead many to flee to this western most part of the end of the world.
By the time Christianity had become the state religion, Eusebius (ca. 325 AD) had recorded countless names of those killed by the state from the time of Tiberius [14 AD], to that of Maximin. [306 AD] Eusebius himself had been placed in prison 309 AD for his Christian faith. Gaul was listed by Eusebius with three centers of church activity at 1) Arles, 2) Lyons, and 3) Vienne. Out of these three centers, much was done to spread Christianity to the Islands.
However, it was those folks from the Egyptian desert that seemed to play the earliest roll in bringing the Christian faith to the Islands. Hermits and monks they were often called. Living a life of strict self-denial, they brought to the Celtic world a life style that would easily be recognized in the wilds of this western most frontier. Besides, who in their right mind would want to come to this part of the world unless their very life depended upon it!
It would have been a slow and gradual process from this migration of monks from Gaul and the Iberian peninsula. These islands would never be the same.
The work of Eusebius is found: "The History of The Church" translated by G.A. Williamson, Dorset Press, 1984. A listing of the martyrs are given on pp. 418-420. The centers of Church activity (Bishoprics) are given on pp. 417-418.