The Old English name was Brittisc or Bryttisc, which in later spelling was Brittish**, misspelt British under the Latin influence of Britannia.
The older name in the 'British' (Brittonic) Languages for their speakers was Brython < Brittones. The Bretons call their language Brezonek, and Cornish was once called Brethonec. All these imply an original *Brittonika.
It is said that the -tt- spelling in Latin appeared around CE 49/50 - a full century after the oldest form in Latin with one -t- as Britann-
** British still rhymes with 'skittish' (two t's) rather than 'whitish' (one t).