There's a perennial debate around Sub-Roman Britain as to whether Rome left Britain or Britain left Rome; although Honorious essentially tells Britannia that it's on its own, this comes after a period of, essentially, increasing British autonomy from the Empire following Magnus Maximus' stripping of garrisons from the North-West in the 380s. If we take Gildas' de Excidio Britanniæ at face value, it's easy to see the mid-5th Century as a period of brutal conquest, as the incoming Anglo-Saxons slaughtered their way through Sub-Roman Britain, leaving its cities and churches in ruins and plunging the country into centuries of Pagan darkness, but that rather apocalyptic vision is most likely quite far from the truth. Rather, Sub-Roman Britain appears to wither on the vine. English conquest aside, archaeology does broadly bear out much of Gildas' underlying narrative: a post-Roman state, or states, briefly achieves some measure of stability and even prosperity, before a series of cataclysmic disasters - an incredibly virulent plague, followed by crop failure and the resurgence of raiding and conquest - leads to political and population collapse and the rise to power of the English.
Archaeological evidence from the fifth century shows a large-scale abandonment of cities in England, a contraction of land use, the periodic reoccupation of pre-Roman fortresses and far more marginal settlement. At the former hillfort of Eddisbury outside Chester, evidence suggests that the area was briefly settled by the Irish, a pattern also seen on the Wirral. This is not a period condusive to the survival of a faith that may not have been heavily entrenched in the first place. In 429, Germanus of Auxerre crossed into Britannia in order to combat the Pelagian heresy that he saw becoming increasingly entrenched there, and purportedly propagated the Cult of St Alban; E.A. Thompson and John Koch suggest that Pelagianism was essentially endemic to 5th Century British Christianity and may have contributed significantly to a 'peasant revolt' which repudiated mainstream Catholicism as much as it did Imperial rule, although this is not a universally held theory. Unlike the early Frankish kings establishing their polities in contemporary Belgica, however, close cooperation with the Church appears not to have been a priority for the incoming English. While their settlements and material culture often mixed closely with the British, the piecemeal nature of settlement and the establishment of many dozens of small, rival polities similarly provided little fertile ground for a Church which was largely tied hierarchies of centralised power and required a basis of education and training for clergy to flourish.
That's not to say that there weren't attempts to convert the English. The first Christian English king, Æthelberht of Kent, was married to the Merovingian princess Aldeberge, daughter of Charibert I and herself a Christian who had brought a chaplain and clerics with her to Kent. Conversion in this period was, however, a highly symbolic act, and one which the English themselves would soon use as a show of political power. Had Æthelberht converted at Aldeberge's request, it was likely that Charibert would have stood as his Godfather, a symbolic position which nontheless would have acted as a sign of Merovingian political dominance over Kent. By converting directly to a papal mission; Æthelberht was essentially able to convert on his own terms, without any symbolic political submission to territorial rivals.
In order to answer this question we must evaluate the time period between the fall of Rome and the ascendancy of Christianity in the Anglo Saxon Kingdoms.
What we first must understand is the clerical structure of the Roman occupation of Britain. While the enforcement of Christianity waxed and waned throughout the period of occupation (roughly the 1st through 5th centuries), there are a few key landmarks that we can establish. The Diocletian Edicts were not thoroughly enforced in the region, allowing many Christian inhabitants (mostly immigrants from the continent) to maintain relative freedom and safety (at least in comparison to their counterparts closer to Rome). In 313 the Edicts of Milan were established, formally permitting the practice of Christianity in the Empire (at least the Western half that was controlled by that portion of the tetrarchy).The spread of Christianity and its continued presence in the region rested on the shoulders of three men. They were Eborius from the city of Eboracum (York); Restitutus from the city of Londinium (London); and Adelfius, the location of whose see is uncertain. The presence of these three bishops indicates that by the early fourth century, the British Christian community was already both organised on a regional basis, and had a distinct episcopal hierarchy. However the struggles to establish Christianity were far from over. The constant ebb and flow of various migratory peoples as well as the presence of many pagan tribes (the Picts come to mind for example), mean that Christianity was far from the law of the land. In 396, Victricius of Rouen was asked to go to Britain to resolve some doctrinal matters. In his De Laude Sanctorum (On the Praise of the Saints), he describes Britain as a wild and hostile place dealing with heresy and paganism. This was the context in which we find ourselves before the withdrawl and fall of the Roman presence in Britain.
In the later portions of the 5th century as the empire crumbled, migration from the continent by Germanic tribes increased exponentially. We see this in the first Anglo Saxon rebellion of 442 as well as the later insurrections which led to the capture all prominent former Roman cities (although it should be noted here that "captured" is a fairly loose term given that there was not truly any established overarching presence of the region, "unified" might be another term that could be employed). While Christianity survived, its power was significantly diminished, watered down by the pagan ideologies of the Germanic tribes that now held power in the region.
With the fall of the empire the Church became an ever more important source of power in the continent, and continued to send missionaries to spread the gospel. The revival of Christianity in Anglo Saxon England therefore began with the Æthelberht of Kent, who held the most power in the region at the time. Aethelberht established ties with the Franks through marriage to a Frankish princess, on the condition that she was permitted to practice her religion (that of the continent), and thus Christianity once again found its place in the highest courts of power in England. Pope Gregory then sent missionaries to Kent in the early 7th century, where the church became more established, with various members in Aethelberht's family also converting at his behest. From there Christianity gradually resurfaced, with its own flair slightly different from the continent (called Celtic Christianity). The entire story of the Christianization of Anglo Saxon England is long and difficult to tell, and is tangential for your question so I will leave a couple of reading suggestions, but not address that portion of English history at the moment.
Now that I have recalled the facts, I think there are a few important details to the story that may address your questions about Christianity in England and in the continent in general. During the period of the 1st-5th centuries, there was an immense expansion and contraction of the Roman empire. The infrastructure that allowed for such a spread did not mean that all those who were conquered had a unifying culture or ideology. The same was true (if not more so) with Christianity, which was in its nascent stages of development. Emperors flipped and flopped between the pagan ideas of their ancestors and the new religion of Christianity in Rome. Christianity at this time was also no where near the religion that we might think of today, with immense flexibility given in the doctrine for the sake of expansion and easy conversions for people of all types. All of this is to say that one mistake that could lead to your confusion is an attempt to transpose modern ideas of nation states, religion and communication among generations onto this time period. It is not until we have an establishment of kingdoms and somewhat stable rule and population that Christianity begins to become a permanent fixture in Anglo Saxon life. Even 5 centuries later (ending with the Norman conquest) the Anglo Saxon Christian kings struggled to establish and maintain a Christian state with the onslaught of Danish and Norse attempts to capture territory and wealth. I would point to the pseudo-state of the Danelaw as an example of this. I would also add that the question of how close the doctrine was adhered to, as well as what modifications were made is unknowable (for example, a Danish ruler might call themselves Christian to establish trust in the region and among his peers, but whether or not he adhered to it is another question). I could soliloquize about this topic extensively so do not hesitate to ask questions.
Sources and Reading
EDIT: The wife of Aethelbreht (Aldeberge) was permitted to maintain her religion, Aethelbreht converted later, likely influenced by his wife. Attribution to /u/BRIStoneman