Always good to take Gildas, Bede, and a lot of Arturian history with a grain of salt. However, it does appear that:
- Some of the earliest Saxons were invited as mercenaries or a counter-balancing force
- Then Saxons, Angles, Jutes, etc came in repeated waves.
- The Britons had success for a period of time beating them back but were eventually overwhelmed, probably due to disunion.
So, the most likely way to achieve this is for a strong Briton leader (Arthur or otherwise) to emerge and build a stronger central kingdom/force that would not only have success in beating the Saxons back, but also have staying power to maintain those gains and survive changes in leadership.
Even so, I don't think the Britons were strong enough to totally keep the Saxons off the island, but they could potentially have kept them constrained to a relatively small portion of England until the waves of Germanic and Norse invaders subsided. If they do that, then they might eventually conquer/absorb those Saxon areas so it's still a Briton kingdom.
If England remains essentially a Briton kingdom, then it would likely retain more Roman culture, learning, and tradition, though it would have a strong Briton flavor to it. They would have a different take on government, society, and religion than the Saxons. They'd likely be Christian, but not sure how Catholic they'd be.
Assuming all that, it's unclear whether/how the contention with Normandy would evolve and whether there's a Norman invasion.