The Northern King Arthur theory is one supported by some Celtic scholars. This theory takes its basis in an historical King Arthur from Historia Brittonum – it sees him as a late 5th/early 6th-century warrior who was leading the fight against the invading Saxons. It then uses the regional bias of the earliest Arthurian sources to argue that these sources imply that this Arthur was originally a hero of Y Gogledd, the ‘Old North’ (that is northern England and southern Scotland), and that his later fame throughout Britain was a later secondary development of this legend.
The Arthurian reference in Y Gododdin is seen as significant in this theory, as there are three or four early (c.550-650AD) ‘Arthur’ names in the ‘Old North’(including a prince of the royal house of Dalriada).
If this ‘Northern King Arthur’ is associated with areas of the ‘Old North’ such as Rheged or Gododdin, then we have to assume that either Badon was not in the south, or Badon was not originally fought by King Arthur.
If however King Arthur had lived somewhere like the East Riding of Yorkshire and possibly York, then his would put Arthur far enough south to fight 5th-century Anglian invaders and it is reasonably close to the most northerly of the candidates for Badon. Then, when this area was lost to the invaders, the traditions of a great defender might have been passed northwards to the surviving ‘Old North’ kingdoms
Personally, I do not think there is a great case made for King Arthur having lived in Northern England.