The three Saiyans forget about Earth, at least for a few years, since they don't know about Dragon Balls.
They keep Gohan hidden from Freeza, because Freeza would kill him if he found out about his potential, fearing that he might be the Super Saiyan.
After some elementary training, they take Gohan with them to whatever planet they wanted to conquer and, with his help, manage to subjugate it.
Over the following years, they secretly train Gohan to turn him into a Super Saiyan. Gohan is brainwashed, forgets about his family and grows up learning about his ancestry. In his mind, his guardians implant hatred of Freeza, a sense of duty and the idea that he is the chosen one to fulfill the prophecy and restore the race to their former glory.
Meanwhile, on Earth, Chichi holds Goku responsible for everything and asks for a divorce. Goku cannot get over the remorse that his wife makes him feel and loses interest in life. Piccolo exploits this and eventually manages to kill him. Then he takes over the world and starts training the next generation of warriors, in case the Saiyans are back.
The Z Humans hate Piccolo but deduce they have to join his army to protect the Earth. Eventually, Piccolo grows different from his father and becomes a proper monarch (albeit an oppressive one) rather than someone who just wants to destroy for the F of it.
How many years pass before the Saiyans are back? What kind of person has Gohan evolved into? Does he even remember that he used to live here once and had a father called Goku and a mother called Chichi? Will his guardians hide him successfully long enough for him to grow stronger than Freeza before the latter finds out? Captain, you take it from here.
