No body has ever proved a motive for Jeremy being the killer.
He had a nice lifestyle, he had an elderly grandmother who he was going to inherit from and eventually through inheritance he would be wealthy anyway
But his life was always about himself: how he had accepted for years his own inadequacy as Sheila herself struggled with her identity, how he failed academically at school, how he failed to make new friends, how he struggled to receive the slightest compliment from his parents, then suddenly a growth spurt and he began to get noticed-he the nonentity suddenly became popular with the opposite sex, how he bragged about his parents' wealth attracted girls, attracted Brett, how he spread the newspapers on the floor carpet of Bourtree Cottage because finally somebody was paying him attention, albeit in the most gruesome of circumstances.
Why did he kill? He took a calculated risk that he almost succeeded with: his parents' secrecy concerning Sheila's illness meant nobody really knew her, his aversion to sharing his inheritance, the dislike of the twins, and the realization that however rich his parents had become they weren't particularly in a position to enjoy their wealth: Nevill working all hours God sent had become haggard, whilst June concerned herself with charitable works without to the detriment of family, struggling with her own demons as she attempted to reconcile her own religious convictions with the hedonism of the modern age.