The Ostrogothic kings in Italy (who took replaced the western emperor as rulers of Rome in 476) were Christian, as were the Visigoths in S. France and Spain. Most of the barbarians serving in the Roman army (including the Goths and Franks) were Christian by the fifth century, though many of them subscribed to a heretical non-Nicene branch of the faith associated with Arius (historians continue to debate what, precisely, this heretical Christianity was, and what it meant for how these people fit in with the larger Christian community).

The first western barbarian king to adopt orthodox Nicene Christianity (the Christianity of the church hierarchy) was Clovis of the Franks, in the late fifth century. The Visigoths in Spain converted to othodox Christianity in the seventh century. The Ostrogothic (Italy) and Vandal (N. Africa) kingdoms were stomped out by Justinian in the mid sixth century before they had made any moves to give up their Arian / non-Nicene Christianity.

Anglo-Saxon England was an outlier; it's not clear precisely what happened with religion in Britain after the Romans left c. 410, but most historians accept Bede's account (written in the 730s, so a few centuries after the fact) in which the island reverts / converts to a paganism rooted in some sort of blend of Roman, Celtic, and/or Germanic folk beliefs (details are sketchy). It's not entirely clear how polemical Bede was being (were things really as pagan as he said, or was he trying to paint his Celtic Christian rivals as pseudo-pagans?), and the archaeological evidence doesn't show as clear a picture of pagan religious practice as we used to think it did (many things like burial with weapons were once thought to be expressions of pagan beliefs in Valhalla, but that view has been pretty thoroughly rejected and we now know that many Christians practices similar funeral rituals in Francia). And we have a Christian sermon (in Latin, no less) surviving from Wales in the 6th century, so Christianity was still alive in some form there after Rome left. It's likely that England had a mix of beliefs (there's a bit of archaeological evidence for Christianity continuing in parts of the South East, and this would be consistent with Bede).

But whatever these beliefs were, the king of Kent, Aethelberht, made a big show of accepting a monastic mission from Rome and 'converting' to Christianity just before AD 600 (though, it should be said, 'conversion' in this period could refer to anything on a spectrum between abandoning paganism for Jesus to committing to live your existing Christian faith with greater dedication). So I'm not sure I completely trust that Aethelberht of Kent was 100% pagan before the missionaries came. His wife, afterall, was a Christian from Francia, so it's not like people in Kent didn't already know about Christianity. It's most likely that, like the contemporary Anglian king Raedwall whom Bede describes (the possible occupant of the famous Sutton Hoo burial), many Anglo-Saxon rulers worshipped Jesus alongside other deities, because hedging your bets is always a smart thing to do when you rule a diverse people and want to be sure everyone's god is doing their part to keep your kingdom afloat.

By the end of the seventh century, though, most of England was identifying itself as Christian.

Charlemagne comes onto the scene around AD800, after all these previous developments were well completed. He made some important reforms to Christian Francia, but the region he ruled had been Christian without interruption since it was part of the Roman empire. Charlemagne sought to standardize Christian practice, brought the Franks into a closer relationship with the Bishop of Rome (by conquering part of Italy), used Christianity to justify his violence against his neighbors to the north (the non-Christian Saxons, in Germany), and revived / created an educational system rooted in Christian + classical principles (in which he was assisted by monks from an also already Christian Anglo-Saxon English kingdom). But he was building on what had by then become a long tradition of western European rulers affiliating themselves with the faith of the Roman empire.

Edit: if you want a good overview of this process, a classic survey text is Peter Brown's Rise of Western Christendom.

That would depend what you mean by "state religion" and, indeed, by "state". The Anglo-Saxons converted to Christianity beginning in the 6th century, and by the death of Penda of Mercia in 655, all Anglo-Saxon Kings of any real political/military significance were confessors and supporters of the church. Now, the concept of "state religion" did not exist, as such. Nor, indeed, the concept of state in the modern sense. But the Christian clergy were closely integrated into the political power structures of the preeminent kingdoms, by the late 7th century, and so the sacerdotal representation and affiliation of the English kingdoms were absolutely explicit. This was, naturally, long before the birth of Charlemagne.