Goku helps the three Saiyans conquer the Heloites other planet while, on Earth, the 100 humans are brought
back with dragonballs. Fearing the Saiyans might return someday (Raditz had said they would, though it's arguable
whether he was telling the truth or just bluffing, considering Vegeta later in canon said they would've never bothered
had it not been for dragonballs), Piccolo trains Gohan and humans train with Kami as in canon.
Eventually, Goku learns about Saiyans history and Freeza. Wanting Freeza to pay for his crimes, he reveals to the
three Saiyans the existence of dragonballs on Earth. By this point, Goku's power is between Raditzes and Nappa's
and the other three Saiyans have made insignificant gains.
The Saiyans return to Earth and Goku tries to convince the others to use a wish to make Vegeta immortal,
acknowledging that this is their only hope to beat Freeza and Vegeta will be a lesser evil.
Vegeta becomes immortal, but this is not enough to just brainlessly go and challenge Freeza. He decides to stay
on Earth for the time being until he finds a way (either by getting stronger or just waiting for Freeza to die of
natural causes). Eventually, he meets Bulma and they get romantically involved in the same way as in canon.
Years go by and Vegeta isn't able to make any significant gains (in canon, he didn't know about zenkais before
getting one against Goku, so he hasn't thought of abusing them). However, this doesn't seem to annoy him; he
has an eternity to wait and has got accustomed to Earth life and the joys of marriage by now. Nappa and Raditz
also have become Z fighters.
One day, Future Trunks shows up, warns them about Androids and reveals the SSJ secret to them. Vegeta finds
out about zenkais. Over the following 3 years, he and Goku spam them like hell and reach SSJ level. Nappa and
Raditz have less potential and top out at the hundreds of thousands. Piccolo is his canon post King Kai self. Humans
are at the tens of thousands. Gohan rivals Nappa/Raditz, impressing everyone.
In my headcanon, Goku caught the virus in the hospital, so he does not get sick in this story. 3 years later, two
scenarios are possible...
1. Gero and Nineteen are defeated without 17/18 waking up. When Cell shows up, he goes to the lab, absorbs
the androids and destroys Earth.
2.Seventeen/Eighteen wake up. The saga is about the same until Cell Games, where Cell destroys Earth when
he blows up, because Goku has no instant transmission.
In either scenario, it's hard to tell what happens to Vegeta, who is immortal. Maybe he gets absorbed by Cell
(assuming Cell can still absorb after gaining perfection), maybe he and Cell part ways and Vegeta goes to rule
Icers empire, maybe Cell keeps him as a sparring partner.
