“I’m so glad you appeared,” she said, looking earnestly into his face. “I was beginning to get worried.”
 
“That’s all right,” Billy answered brightly. “You mustn’t worry about me.” He put his suitcase on the chair and started to open it.

This exchange occurs after the landlady has shown Billy to his room in her bed and breakfast. It reveals Billy to be a positive but somewhat clueless person. The landlady is clearly not referring to Billy’s safety when she says she was starting to get worried, but Billy completely misses this hint. His practicality and industriousness make him so focused on himself and the task at hand, that he doesn’t realize that something is off. 

“Now wait a minute,” he said. “Wait just a minute. Mulholland ... Christopher Mulholland ... wasn’t that the name of the Eton schoolboy who was on a walking-tour through the West Country, and then all of a sudden …”

“Milk?” she said. “And sugar?”

“Yes, please. And then all of a sudden ...”

This conversation happens in the living room of the landlady’s bed and breakfast as Billy tries to recall where he has heard the name Christopher Mulholland before. It again reveals that he is practical to a fault. Instead of making the connection between the disappearance of the boy and the landlady in front of him, he seems intent on remembering the details just for the sake of remembering them. As a result, he does something he should probably by now realize is a dangerous mistake: he accepts the tea prepared by the landlady.