This subject is actually filled with question marks, and is really only recently being examined critically.
The original thinking about what happened in North Africa was, as talked about by Gibbon:
"The Northern coast of Africa is the only land in which the light of the gospel, after a long and perfect establishment, has been totally extinguished... Within fifty years after the expulsion of the Greeks, a lieutenant of Africa informed the caliph that the tribute of the infidels was abolished by their conversion."
Despite Gibbon's love of dramatic phrasing, it's generally accepted the idea that the Christians of North Africa converted to Islam, as opposed to some kinda mass blood and slaughter.
However, the speed and scope is the big question. For example, it's not known how exactly big the Christian population was in North Africa at that time. Obviously there were very important cities that were heavily Christianized, like Hippo, home of Saint Augustine, but the area was also filled a complex mixture of indigenous Berber, Romanized Berber, and Byzantine populations "within a complex mosaic of varying social, economic, religious, and political forms" (Boone 52). Once you add in an Arab military garrison that was 4-5 times the size of the equivalent former Byzantine military garrison, that identity and identity transition gets more complicated. There's excavations of Islamic style houses that were built over a former Roman bath, but those same houses also had aspects that seemed roman, like central courtyards. There are a lot of other artifacts being excavated from the period, but like the problem with post-roman anglo-saxon england, it's not always possible to tell whether an islamic ceramic is an indicator of the person using it's islamic identity, or it was a romanized berber who happened to be using islamic products.
What does seem clear, is that there seems to have been a failure to develop a coherent "alternate" Roman identity to the Arabs for Romanity or Christianity to continue forward, unlike with say the Copts in Egypt. This might be because if the Christian population was primarily urban (unlike say a possibly non-Christian Berber countryside), then the breakdown in North African trade (which underpinned those cities), would fundamentally change how those residents survived, shifting their interests from the purely commercial to securing patronage from the new Arab garrison elite, with the best way to do that being conversion.
Which meant until the rise of stronger regional identities, the ones that survived were either the centralized Islamic Arab ones at the top, or the hyperlocal berber tribes at the bottom. There simply wasn't a place for the urbanized Christian Roman identity of the Byzantine past to linger on for long in the mix.
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