In the novel One, Two, Buckle My Shoe, Reilly is a dentist at the dental practice of 58 Queen Charlotte Street and a colleague and partner of Henry Morley. Morley considers Reilly to be a very able dentist, with first class diplomas and thoroughly up to date. However Morley's sister Georgina thinks that Reilly's hand shakes and that he drinks too much. It appears that Morley had hinted to Reilly on several occasions about his drinking.
After the murder of Henry Morley, Reilly is for a time considered a suspect but not seriously. He had opportunity because his office was near Morley's but the motive is weak. Reilly is said to be hot-tempered and likes arguments particularly over politics but these disputes never went very far. As for the idea that Morley had committed suicide after injecting too much anaesthetic into Amberiotis and cuasing his death, Reilly dismisses it as impossible to do by accident--to inject that amount of drug that the coroner founnd, Morley would have to be drunk or have intended to kill him deliberately.
In the later part of the book, Poirot spots Reilly at a shipping office booking a passage to America. He intended to go there and not return. It appears that he he run up severe debts (presumably from his drinking). As for abandoning his practice, he tells Poirot that it is more like his practice is abandoning him.
Reilly is described as a tall, dark young man, with a plume of hair that fell untidily over his forehead. He had an attractive voice and a very shrewd eye.