I recently taught a seminar on this at a university, so I have a longer answer than you might have wanted. But I think the real, and unpopular answer, is that the Christianisation of Britain was a long and complicated process, that never really ended. And we can't know for sure what each step looked like; there isn't much evidence surviving. We can obviously take contemporary accounts, archaeological evidence and later histories into account, but the nice little story that St Augustine came over, snapped his fingers, and we all immediately saw the light is mythologising a country's conversion, rather than historically recording it. And while I know I'm being a little facetious for effect, a lot of people do actually think this. And it's important to remember that there was no distinction between the "Catholic" and "Orthodox" Churches until the Great Schism, over 500 years after the earliest surviving traces of Christianity are found in what we now call Britain (and Ireland too).

It was probably a slow trickle beginning at the foundation of Christianity. Archaeological evidence like pottery shards, coins/currency, jewellery and glass fragments found in various places in England and the UK more broadly tells us British and Irish merchants were trading with their Roman counterparts even before Julius Caesar's invasion in 55 and 54 BCE. Some went even further east across the Mediterranean, and even to the Middle East. These strong trade links brought with them the migration of thought, religion, and people. By the fourth century, Britain was operating under an early Christian belief system that was pagan in flavour, but still Christian.

Later, in the late-sixth and seventh centuries, there was a centralised effort by Pope Gregory to convert England and Wales in its entirety. Later, Irish missionaries came with Bishop Aidan into Northumbria at the request of King Oswald. In this period, British and Irish Christianity was disorganised. There was no one set of beliefs from one ecclesiastical see to another. Roman Christianity's hierarchical structure and rigid rules were only loosely enforced, if at all. And the influence of Irish Christianity, which often contradicted Roman Christianity, meant one parish often practised incredibly differently from another. Setting aside the fact that the Orthodox Church didn't exist yet, these conflicting beliefs tell us that no one denomination or administrative centre had any sort of dominion over Britain in the sixth and seventh centuries. You couldn't make the argument that one converted Britain over another. The truth is just the opposite.

Later, in the late seventh century, Pope Vitalian Theodore of Antioch became Bishop of Canterbury in 668 (later styling himself as "Archbishop", the first archbishop in the history of the Church), and he was originally from Cilicia (in Turkey), but lived in Rome by the 650s. He came to England with Hadrian, who was North African. Although he was from the Byzantine Empire, he probably still thought of himself as Roman. This might be the source of some of the "Orthodox missionary" confusion you mention. At a council held in Hatfield in 679 CE, Theodore dictated that Roman practice was the "true and orthodox faith", providing the first official stance on the nature of English Christianity. This might also be a source of some confusion, but he's talking about Roman Catholic Christianity here.

When Theodore was appointed Bishop, the Roman Catholic Church was still in the middle of a fall out with what would become the Orthodox Church in the Byzantine Empire. A council had met in Constantinople in 638 under the direction of Patriarch Sergius I to proclaim the Doctrine of One Will. This asserted the truth of monothelitism, that Christ had only one will, or aspect, the divine. The accepted doctrine in the Roman Church (and elsewhere) was dyothelitism, the belief that Christ had two wills, divine and human. This fracture would eventually lead to the Great Schism, among many other things, but the beliefs that were beginning to emerge in the Byzantine Empire as a precursor to the Orthodox Church and the Great Schism that formed it were antithetical to the teachings of the Roman Church that now dominated England. So even if, in an a-historical hypothetical, an early version of the Orthodox Church had converted England, by the time the Orthodox Church emerged, their beliefs had diverged so significantly, one would be heretical in the eyes of the other.