Since you've got that second paragraph in your description, I can expand a bit on earlier Medieval conversion, specifically the Anglo-Saxon conversion period.

The difficulty with medieval conversion processes is that the written sources for them by and large come from a Christian perspective; for Anglo-Saxon England that is largely Bede and early hagiographies, especially St. Wilfrid's. This is compounded by the fact that these are not contemporary (although the Vita Sancti Wilfridi is probably by a follower of Wilfrid's and is near-contemporary). So we have these written sources, written and disseminated after the fact, trying to promote Christianity and its expansion. Despite these problems, it is from these written sources that we get the idea of kings converting, and subjects following suit. Considering this, and taking a look not only at those sources but also archaeological and material evidence, the historical consensus has moved away from the idea of a king’s primacy in the process. More recent work highlights instead a longer undercurrent of Christianity before conversion and paganism afterwards. While a king may have converted fairly early in the overall transition of a region, Christianity was probably already established to some degree. In this sense Anglo-Saxon England is also a good example, since until essentially the last 25 years or so it was thought that Christianity had almost entirely been abandoned after the withdrawal of Roman power, a stance that is increasingly being revised.

Bede’s Historia Ecclesiastica Gentis Anglorum, written in 731, is a good case in point for exploring this. Bede’s history is on the English Church foremost, and although the text is used as a source for the early Anglo-Saxon period more broadly, that distinction is emblematic of the Christian promotion mentioned. The work is structured to show a steady growth and expansion (with some early setbacks) of Christianity, although ultimately Bede ends on a more ominous note. The growth for Bede took place through watershed moments involving major figures and events, such as rulers and bishops or synods and battles. Unsurprisingly pivotal to this expansion then is the conversion of kings, and there are two main scenes in the text.

The conversion of Aethelberht of Kent comes fairly early in the book. Although Bede allows that Aethelberht married a Christian Frank who had brought priests with her, the impetus is on Aethelberht and the missionary Augustine. Augustine meets with Aethelberht outside (according to Bede a pagan practice out of fear of magic), is allowed to establish a church at Canterbury, and goes on to convert Kent in the early years of the seventh century. Notably, the Frank Bertha apparently established a small congregation at St. Martin's outside Canterbury.

Later, Bede recounts the conversion of Edwin of Northumbria by the missionary Paulinus (mid-seventh century). In this case, Edwin wants to marry a (now-Christian) Kentish princess, and is required to convert first. So, Paulinus goes to the king for some time, and after a dramatic rejection of paganism by Edwin's own high priest, the king converts along with his retinue and implicitly wider Northumbria. Bede, a Northumbrian himself, gave a lot of emphasis to Edwin and his Christian successors, although from reading the text the contemporary pagan king Penda of Mercia seems like an ostensibly more successful leader (twice defeating the Northumbrians along with subjecting several other groups). While there is no reason to doubt that Paulinus converted Edwin, placing this as the key moment in Northumbrian church history is not emphasising historical truth. Rather both it and the conversion of Aethelberht are participating in a genre of rulers' conversions in Christian writing.

The genre, at its core, boils down to the shadow the Emperor Constantine had on concepts of good Christian rulers in the Early Middle Ages. Constantine (according to his own biographers such as Eusebius) had a vision of Christ, followed by a sign in battle. He went on to convert, became a measured and good Christian ruler, and attempted to both expand the church and resolve disputes. This idea, of a ruler converting on going on to further greatness was expanded by Gregory of Tours, when describing the Frankish king Clovis. In Bede’s case, the idea is similar, there is disenchantment with the status quo, a revelatory moment, and mass conversion and greatness to follow. Certain symbols also carry through, such as a vision of the cross which Bede asserts a slightly later king Oswald to have seen and used in battle - like Constantine. To be clear, these stories are not restricted to these instances, and can be seen in Roman successor states across Continental Europe (another good example is King Reccared of Visigothic Spain).

Triumphal at these times, at other moments Bede gives what seems to be a more revealing look at Christianity as it spread in Anglo-Saxon England. A king of East Anglia, Raedwald, is described as worshipping both pagan gods and Christ with two adjacent altars. In several other instances, times of misfortune such as defeat or disease result in reversions to paganism. There are also brief mentions in passing that paint a more complex picture of the conversion process beyond the elite. Bishop Cuthbert, for example, was preaching in Northumbrian hinterlands where beliefs in magic apparently persisted. Throughout this there is a tolerance of pagan practice and ideas: in a letter from Pope Gregory to Augustine for example Gregory emphasises that Christianity should be as accommodating to pagans as possible to encourage conversion, and that places of worship can be reused. Although Bede would like to skirt around this elephant in the room, the reality was that conversion was complex both before and after the individual conversion of kings. This is the picture we can see from the archaeology. The most comprehensive and measured study of Anglo-Saxon burial to date, by Hines and Bayliss in 2013, recast traditional ideas about Christianity’s impact. Formerly, there was a “Final Phase” or “Conversion Period” to burials, where pagan furnished goods and cremations declined and ultimately east-west unfurnished burials declined. As the name implies, this was tied to the arrival of Augustine. Hines and Bayliss have shown, though, that this decline started well before Augustine’s arrival in 597, and was complete before the conversion of England is typically post-dated around 680. Other studies have shown the continuity of Christian communities from the Roman period, especially developed in Worcestershire, Kent, and at St. Alban’s. A similar revision has taken place with secular written sources. Law codes were traditionally seen as a concept brought by Christianity, but Lisi Oliver has shown that the earliest - the Laws of Aethelberht (the same king as mentioned above) - pre-date Augustine at least in an oral tradition, and Nicholas Brooks has even suggested that it existed in a runic format.

In brief, then, Christianity and conversion as concepts are already more complicated than the actions of kings in written sources belies. It is perhaps better to see conversion as a spectrum even, with pagans and pagan practices slowly being phased out in favour of Christianity. A few objects from the seventh century can highlight this. Firstly is the pectoral cross of St. Cuthbert (once again aforementioned). This large silver and jewelled cross was buried with Cuthbert, along with several other small grave goods such as a comb. Despite these objects, the traditional Cuthbertine tradition describes him as one of the greatest Christian ascetics. Owning a valuable cross shows a more complex man, who still at least entertained pagan notions about the importance of grave goods (or his monastic community did when burying him). Another is the Frank’s Casket. This small chest draws explicit comparisons between Biblical stories and pagan Germanic stories, depicted on the panels of the box. Both this and the cross are ornate pieces of art, that would take great skill and considerable resources to produce. That people within elite circles did not have a black and white conception of Christianity versus paganism implies a level of tolerance for pagan practices. As such, hard-line terms such as pagans or Christians are to a considerable degree retrospective constructions. Beyond difficulties reconciling Christianity and non-Christianity, there was disagreement within the church itself, such as the Easter debates between Irish and Continental clergy in England. It is perhaps notable that there is no evidence of violent reprisals or persecutions against pagans in early Anglo-Saxon England. While we’ll never know exactly how contemporaries thought, the picture presented is one of some synthesis and (albeit begrudgingly by some like Bede) a degree of tolerance that faded over time.

In terms of books, I'd strongly recommend Chris Wickham's The Inheritance of Rome, which is just great in general for an influential idea of Western Europe after the Roman Empire. Other good texts in my opinion include the chapters on the church in the Oxford Handbook of Anglo-Saxon Archaeology as well as the chapters on religion in the new Cambridge Medieval History. At the end of the day, though, dig into some primary sources! Penguin has a good translation of Bede's History and if you can give Eddius Stephanus' Vita Sancti Wilfridi a read. Another good one that comes to mind is the Vita Sancti Germani by Constantius of Lyon.

Further, some of the other works I was referencing here:

J. Hines, A. Bayliss, Anglo Saxon Graves and Grave Goods

L. Oliver The Beginnings of Anglo Saxon Law

I know I didn't answer your direct question, but hopefully this is informative!