Answer:
After giving the apple to Aslan, Digory feels more content, peaceful. C. S. Lewis wrote the Chronicles of Narnia to be Christian fiction therefore this scene symbolizes giving up temptation(s) to God. Digory was tempted to take the apple to heal his mother, to eat the apple, to take part of the apple to his mother. He was relieved of all this temptation when he gave it to Aslan, who let him take one from the new tree that sprouted from the one Digory had brought back to heal his mother.
It would be "traverse the lane in record time"
I'm afraid this question cannot be answered without us being able to view the excerpt mentioned in the question.
