The period 600 - 700 is the period when it is being defined. There are a few dozen anglo saxon kingdoms, places like Spalda, Wixna and Gyrwa and several british kingdoms such as Elmet and Dunotig possibly Lindisfarona and The Peak in what is now England at the start of the 7th century. The major anglo saxon kingdoms start to become dominant by the the start of the 8th century however and the number of british kingdoms have contracted further. But Britons continue to exist after 700 even in anglo saxon kingdoms kingdoms like Wessex. British landholders have a lower status than their anglo saxon counterparts, a british noble, ie someone who holds 5 hides has a wergeld of 600 shillings whereas his AS counterpart has a wergeld of 1200 shillings, but the british landholder is still much higher status than an AS holder of 1 hide, ie 200 shillings who in turn is of higher status than a briton with 1 hide who has a wergeld of 120 shillings. Given that all the above have duties and obligations to landowners with larger estates and that these landowners have many legal and economic ties with the church, those who identify as anglo saxon in dealings both within the church and outside the church will have a better time of it. Most britons attempted to identify as anglo saxon but as Alex Woolf points points out, if firm legal distinctions existed between Britons and Anglo-Saxons, individuals and communities will not have slipped from one identity to another with ease. Indeed, there are strong economic incentives to preserve this segregation when viewed from an English perspective. If you didn't go to church, you had no chance at all, though it still didn't guarantee them success. The stones inscribed in Brittonic now located at Lady St Mary Church in Wareham are dated between the 7th and 9th centuries. The Church was a high status church, reputedly the place of burial for King Brihtric in 802, so here is one example of britons of some status attempting to elevate their positions by becoming anglo saxons. Even though they were never accepted as such, you can imagine that the peasants who worked their lands were told, by them, to speak english. Anyone who went around saying that the church was wrong about the date of Easter was likely to get fined to such an extent that they'd probably end up backrupt.But it was most likely a pretty small factor, which only really influenced the language spoken by the clergy and the written language, far from being an indicator of general use. The division between Britons and Anglo-Saxon seemed to have been already defined.
I don't know that they did run in parallel in the 8th century. The Ionan monks at Ripon were expelled before 664 and when Oswiu decided at Whitby to follow Rome, Colmán the Bishop of Norumbria and the Ionan supporters who did not change their practices withdrew to Iona. They were allowed to take back some relics of Aidan. Wilfred, an advocate of Rome, became Bishop of Northumbria and he filled all the newly vacated positions with his men, ironically, clerics from ireland who agreed with the position of Rome. They even translated Bede's latin into english. No one was translating anything into brittonic.Yes the Church was very powerful but at the same time we are not really speaking of 2 coherent competing institutions, on one side we have the Roman Catholic Church supported by a continent wide network and the strongest West European state on the other we have an array of local monasteries and churches with some shared traditions but without the institutional power to even necessarily create an identity around it.