It's complicated, and the qualifier of 'English' makes things even fuzzier, because Arthur has a fairly long history before he appears in anything that could be called English. Whether the folklore around him stems from an actual flesh-and-blood person is still argued today. Peter Field, a major Arthurian scholar, very recently came out with his theory on exactly where the original Camelot was, while other historians will just say "Geoffrey of Monmouth made it up," full stop. But your question was more about whether medieval people believed in him.
Gildas (6th century) gives us the first mention of a hint of Arthur. In his pretty morose De Excidio Et Conquestu Britanniae, he ruefully mentions the Battle of Badon Hill, something that would appear later in Arthurian writings, but it's totally possible that the event was just picked up by later writers and given to Arthur. Y Gododdin is (arguably) the next source to mention him possibly as early as the 7th century, but it doesn't specify whether the author thought he was real or not, saying of the hero Gwawrddur, "He fed black ravens on the rampart of a fortress / Though he was no Arthur." In the 9th century, Nennius's Historium Brittonum gives us Arthur in the flesh, an actual hero with a list of great battles fought as part of an attempt to write a 'national history' for Britain, with all the elaboration and mythic origins that entails. The Historium traces the legacy of the British people all the way to Brutus, a refugee from Troy who settled in Britain after stopping by Rome. It's a Welsh monk writing a legendary history with a sense of nationalism. But Nennius's thunder gets stolen by another Welsh writer, the much more famous Geoffrey of Monmouth in the 12th century. His Historia Regum Britanniae is probably the biggest source of what would later turn into the most widespread Arthur tradition, and it's here that credibility gets fuzzy.
Geoffrey claimed he got all his source material from a mysterious book in Welsh. Obviously, he never produced that book because it probably didn't exist, and he ripped most of his material right off of Nennius, but the interesting part is that the Historia got extremely popular, and some of his contemporaries openly doubted how legitimate of a historian Geoffrey was. To me, at least, the fact that there was room for doubt and controversy implies there were people out there thinking this was indeed plausible stuff. Of course, there's another reason for that--the Historia includes the enigmatic Prophecies of Merlin, which were also a very popular addition that doesn't get as much attention. To the aristocracy, prophecies are extremely useful as political propaganda, but that's another story; the Historium got popular and caught the attention of romance writers across the channel.
French poets like Marie de France thought Arthur was great fun for making up endless poetry that doesn't seem to be taken as history. But Wace, a writer from Jersey, has the clever idea of writing up a romanticized, very Norman version of the whole story of Britain that becomes his Roman de Brut, the Romance of Britain. Like Geoffrey, Wace sidesteps responsibility by saying "I'm just reporting what I hear from Britain." So far, all these sources have been Welsh or French.
To the best of my knowledge, Layamon's Brut (ca. 1200) is the first English source to claim total faith in his work and his version of the Arthur story. Coming right on Wace's heels and using him as his main source, Layamon writes his own chronicle in archaic Middle English, in verse that looks a lot like old Anglo-Saxon form. In his prologue, he introduces himself as a humble priest of Areley Kings who's done a lot of traveling to compile this work, and he swears on his faith that to the best of his knowledge, everything he's written about is true. He take liberties with his history that includes a VERY long Arthuriad, but for the most part, he tries to be a responsible historian, and the chronicle has a sense that it's trying to present itself as the 'authentic native' version of the story. Areley Kings is right near the Welsh border, so Layamon might well have had a Welsh informant helping him along.
The Welsh tradition kind of disowns Arthur after the English get their hands on him. The Arthur of the Mabinogion is a strange, almost funny character who isn't anything like the hero of French romances, and later works like Malory's Morte D'arthur read like pure entertainment. Edward III certainly put on a show to play up Arthurian style, right down to hosting a tournament where he and some of his close friends dressed up like Arthur and knights of the round table, but that doesn't suggest he really believed it so much as he was very good at publicity--and Froissart's chronicle backs him up. Froissart draws a direct comparison between Edward and Arthur, but again, that likely strengthens the idea of Arthur as a character of propaganda rather than showing that someone actually believed in him.
In short: maybe. And if there were true believers, they probably came in the immediate wake of Geoffrey. Layamon wrote in a language that could have been understood by the common people, if anyone cared to read the text out loud, so does that suggest that the commons might have been more likely to believe it? Maybe, but we don't know for sure. All that probably raises more questions than it answers, but I hope that gets you an idea!
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