I can't really speak to Islam, but in the case of the Latin west Christianization was a long and complicated process that arguably was never entirely complete. Situations differed greatly from place to place and from time to time, so I'll hop between a few examples in the hope that a more general picture can emerge.

Obviously, even though the local ruler has converted and the populace is now nominally Christian, pagan practices and worship don't simply stop. Even in areas that one would think Christianity had been long established, such as Italy during the time of St. Benedict or 6th century Byzantium we find references to pagan temples still in active use (cf. Gregory the Great's Dialogues). The conversion of a ruler was generally followed by the promulgation of laws designed to eliminate practices, religious and otherwise, which were considered especially abhorrent to Christians. For example, in Scandinavia the eating of horseflesh (i.e. sacrificial meat) and the exposure of infants and the elderly was banned on the conversion of the various local leaders. As time went on and Christianity consolidated itself, more laws banning pagan practice were enacted. These occasionally caused kickback and the repeated insistence on the need to follow them by various officials indicates that enforcement was not uniform or simple.

Meanwhile, Christian missionaries worked to root out pagan worship on a local level. Often this included doing things like destroying pagan shrines, chopping down sacred trees, challenging pagan wonderworkers, and so on. Unsurprisingly, this could really piss people off and these missionaries regularly faced violent opposition. As an example, Boniface, the most famous of the great Anglo-Saxon missionaries in the Frankish Empire, was murdered along with his companions during a missionary expedition to Frisia. The close ties between the process of Christianization and the imperial project of the Carolingian Empire (and other similar projects, like the campaigns of Olaf Tryggvason in Norway) likewise engendered violent revolt, such as that which precipitated the Bloody Verdict of Verdun, where Charlemagne massacred thousands of Saxons due to their rejection of his rule and the Christianity that came with it.

Even after violent opposition had largely been quelled and "Christendom" had in some sense been established through much of Western Europe, practices were widely variable and local beliefs were often anything but Christian in any sense we'd recognize. There were thus a series of efforts to catechize and reform religious practice that went on throughout the Middle Ages. For instance, the Carolingian's mandate that local priests must be able to say and explain the Lord's Prayer, read the mass, and properly administer baptism, not particularly onerous requirements which point to the bare bones nature of Christianity at the time, especially in rural areas. The greatest of these probably being the massive reforms of the 11th and 12th century, which climaxed in the Fourth Lateran Council which laid out the basic requirements of the Christian life for laypeople. Even still, non-Christian practices did not vanish and there are scattered records in hagiography, law codes, histories, inquisition records (cf. Carlo Ginzburg's The Night Battles), etc. persisting well until the era of the Reformation and beyond.

Really this is a massive topic with tons of places to dig into and explore further, especially considering the extremely local character of Medieval Christianity. The best primary sources to look for more, in my eyes are hagiographies, particularly those of the monastic missionaries (Boniface and his contemporaries are always very interesting), though it may be difficult to discern what is an accurate report of the situation on the ground and hagiographical trope. Secondary sources there are any number depending on when and where you're concentrating, Aron Gurevich has written a bit, the above mentioned Carlo Ginzburg, Valerie Flint on magic, Alan Cameron on Greek and Roman paganism's decline, and many more who are not on the forefront of my mind at the moment.

apologies for lack of links and any typos/awkwardness of phrasing, this was all typed rather hurriedly and I'm happy to clarify anything if necessary.