From what does Framton Nuttel suffer?

Framton Nuttel is supposed to be undergoing a "nerve cure" by residing in the English countryside and avoiding all excitement or exertion. Nowadays he would be diagnosed as "neurotic" and would probably be referred to a psychotherapy for expensive consultations. The author does not describe Framton's symptoms, but they undoubtedly include nervousness, insomnia, nightmares, some degree of paranoia, depression, and apprehension. 

"The doctors agree in ordering me complete rest, an absence of mental excitement, and avoidance of anything in the nature of violent physical exercise," announced Framton, who laboured under the tolerably widespread delusion that total strangers and chance acquaintances are hungry for the least detail of one's ailments and infirmities, their cause and cure. "On the matter of diet they are not so much in agreement," he continued.

The fact that he says, "On the matter of diet they are not so much in agreement," seems to be hinting that he wouldn't mind being invited to stay for dinner. He probably has a good appetite. His bad nerves are probably only imaginary. Mrs. Sappleton shows little interest in his health problems or his rest cure. She already plans to serve him tea and biscuits and hope never to see him again. Her attitude about this visitor reflects Saki's own attitude about weaklings with imaginary ailments. The reader does not have much sympathy with Framton either. This is at least partly because the reader doesn't approve of a complete stranger imposing on families all over the region with nothing to offer but letters of introduction from a woman they can barely remember. There has to be very little reader sympathy for Framton Nuttel in order to make the final scene truly funny.

Framton is made to order as a patsy for Vera's ghost story. He is a good example of how a competent professional fiction writer will create characters to suit his plot. Framton is a nervous wreck who is seeking "complete rest, an absence of mental excitement, and avoidance of anything in the nature of violent physical exercise." Instead he finds himself in a zany household where his hostess appears to be completely insane. He ends up getting plenty of mental excitement and violent physical exercise.

Framton grabbed wildly at his stick and hat; the hall door, the gravel drive, and the front gate were dimly noted stages in his headlong retreat. A cyclist coming along the road had to run into the hedge to avoid imminent collision.

The fact that he can run so fast and so far suggests that there is nothing seriously the matter with him.