The AD122 covers the central part of Hadrian's Wall and is well documented elsewhere on this forum, other local buses cover the eastern end from Corbridge on to Wallsend and the Stagecoach 93 covers the stretch on from Carlisle to Bowness on Solway,
But the stretch between Birdoswald and the outskirts of Carlisle is far more difficult since bus subsidies were withdrawn. Then the basic services that were restarted ceased during Covid and were very slow to restart.
They finally have, and are operated by The Border Rambler (a Community based service). They require careful planning to use, but at least do exist.