alt Hacker Newsafter non-trivial inquiry from far-away California, my best understanding is that the Celts did gracefully embrace the Christian faith among the monks and those serious about religious life. Since there were vivid and lived religious traditions alive at all times through history, this transition was not uneventful. However the kind of "top down" and by-the-sword conversion that did occur e.g. the Baltic tribes, was not the case with the equally fierce Celts
But the graphic suggests that Celtic Christianity was in some sense theologically distinct from Chalcedonian Christianity, and that doesn't seem to have been the case. The main ways that the Christians of Ireland and Britain differed from those of continental western Europe seem to have been in the shape of the monastic tonsure and the calculation of the date of Easter; and in the latter, at least, British and Irish Christians were in conformity with Rome by the end of the eighth century. (There was also an emphasis on penance and absolution as a private rather than public rite, but this was ultimately adopted by the wider church.)
There doesn't seem to have been any doctrinal disputes, nor any suggestion that British and Irish Christianity was in any way separate from the Church of Rome.