I'm aware this scenario may border on ASB, but humor me, please:
Arthur I, nephew of Richard I, leaves a male heir when he dies (he can die on schedule or shortly thereafter). John Lackland has succeeded as king of England, and the other titles he usurped, so Arthur is only holding Brittany - and maybe one or two other French fiefs of the Angevin Empire. This male heir is under the regency of his mother - let's make her Constance of Toulouse or Sancha of Aragon, rather than the French princess Arthur was betrothed to, only so that it is easier for Arthur to leave a child (the French princess was only born in '98).
By treaty (probably sometime before Arthur's death (which is likely to still be blamed on John) which is before the birth of Henry III) Arthur's line is acknowledged as heirs to England after John's kids or in default of their being a Johannine heir.
Now, Arthur's little sprig establishes a cadet branch of the Plantagenets in Brittany. John can't get too heavy handed once this kid is an adult, or else the barons in England won't be inviting the French king, but rather the duke of Brittany over the Channel (but he probably still will, things will escalate and...)
Say this Plantagenet-Anjou branch Brittany lasts about 100-150 years (it's when the House of Montfort took over OTL), how does this influence England's relations with France and vice versa. Might we see a duke of Brittany aspiring to the throne of England? The king of France willing to stir the pot?