when did the arthurian legends reach their height of popularity
With the outbreak of the Second World War.
Modern legend
As Taylor and Brewer have noted, this return to the medieval "chronicle tradition" of Geoffrey of Monmouth and the "Historia Brittonum" is a recent trend which became dominant in Arthurian literature in the years following the outbreak of the Second World War, when Arthur's legendary resistance to Germanic enemies struck a chord in Britain.
12th-century 12th-century. 18th-century
King Arthur King Arthur King Arthur
Introduction Introduction Geoffrey of Monmouth
The legendary Arthur developed as a figure of international interest largely through the popularity of Geoffrey of Monmouth's fanciful and imaginative 12th-century "Historia Regum Britanniae" ("History of the Kings of Britain"). 
Arthur is a central figure in the legends making up the Matter of Britain. The legendary Arthur developed as a figure of international interest largely through the popularity of Geoffrey of Monmouth's fanciful and imaginative 12th-century "Historia Regum Britanniae" ("History of the Kings of Britain"). In some Welsh and Breton tales and poems that date from before this work, Arthur appears either as a great warrior defending Britain from human and supernatural enemies or as a magical figure of folklore, sometimes associated with the Welsh otherworld Annwn. How much of Geoffrey's "Historia" (completed in 1138) was adapted from such ear  The old notion that some of these Welsh versions actually underlie Geoffrey's "Historia", advanced by antiquarians such as the 18th-century Lewis Morris, has long since been discounted in academic circles. As a result of this popularity, Geoffrey's "Historia Regum Britanniae" was enormously influential on the later medieval development of the Arthurian legend. 
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King Arthur [SEP] Modern legend
Retellings and reimaginings of the romance tradition are not the only important aspect of the modern legend of King Arthur. Attempts to portray Arthur as a genuine historical figure of , stripping away the "romance", have also emerged. As Taylor and Brewer have noted, this return to the medieval "chronicle tradition" of Geoffrey of Monmouth and the "Historia Brittonum" is a recent trend which became dominant in Arthurian literature in the years following the outbreak of the Second World War, when Arthur's legendary resistance to Germanic enemies struck a chord in Britain. Clemence Dane's series of radio plays, "The Saviours" (1942), used a historical Arthur to embody the spirit of heroic resistance against desperate odds, and Robert Sherriff's play "The Long Sunset" (1955) saw Arthur rallying Romano-British resistance against the Germanic invaders.