Vegeta was sent to Earth instead of Goku?

[GSM123]

Like Tarble, Vegeta was born a low class warrior and got send to Earth for being a shame to the Vegeta family. Kakarott is born in the low class also but is taken under Raditz’s tutelage instead of being send to Earth.

 

[SSJ2]

So he starts off as strong as Goku did? It's hard to say. Goku only lost his Saiyan instincts due to hitting his head and I doubt Vegeta would do the same. If he retains his Saiyan nature, it might get him into trouble at a young age. There are a plethora of fighters who were capable of one shotting Goku at the BoDB. If Vegeta pushed his luck against any of them, his life would end early. However, if Vegeta is patient and allows himself to train for years, he would easily be capable of destroying Earth's population by using the Oozaru transformation. Knowing Vegeta it seems more likely that he would charge into battle at a young age and would get decimated by any of DBs top tiers.

 

[Papasmurf]

So Vegeta doesn't have his Super Elite powa, but is a low-class instead? He basically just follows Galu's life to a T except that his natural attraction to Bulma would stop him from marrying Chi-Chi. Vegeta ends up spending the night at Bulma's place after becoming world champ in the 23rd and has Trunks much earlier than in canon. I'd assume that due to Vegeta lacking many S-Cells ( ) Trunks isn't born with the power to go SSJ or the insane power his counterpart displays in the series, and so ends up following Gohan's route, except that due to what little brotherly bond Kakarotto and Raditz have, they arrive on Earth simultaneously and kill both Vegeta and Piccolo. They kidnap Trunks to become one of them after this, and without Vegeta to talk him out of fucking Earth women, Nappa decides to make a fuckton of Saiyan babies after killing all the Earthling men. Fearful of a new generation of Saiyans being born, Freeza blows up the Earth and kills all of the Saiyans.

 

[Captain Cadaver]

Without the familial ties, Raditz never recommends going to Earth so the Saiyans never visit it. Without the experiences to help him grow, Goku would probably never surpass Raditz if they continue serving Freeza.

Meanwhile, Vegeta arriving on Earth 5 years before Goku opens up a lot of variables. I'll go the simplistic route and say Gohan just finds him earlier in the same spot, Vegeta hits his head and things go the same with him having most of the traits Goku did. The big difference is that he's 17 when Bulma arrives and likely has had his adult growth spurt, leading to attraction forming between them. He'd also naturally be stronger than Goku was at the time, so the journey for the Dragon Balls goes pretty easy and the group aren't captured by Pilaf. Bulma would decide to hook up with Vegeta, so Yamcha gets his wish of no longer being afraid of women and becomes popular with the ladies. Vegeta still trains with Roshi (though Bulma comes to the island as she did when Yamcha trained and Lunch is never found as a result) and ends up winning the 21st TB. It's at this point he gets traits closer to his canon self as he'd be more arrogant from having proven his strength, though still searches for the Four Star Ball. He solos Muscle Tower without getting too fatigued, manages to kill Blue in the cave with a Kamehameha but still gets beaten by Tao Pai Pai (albeit, being strong enough to still endure a Dodonpa even if the ball isn't at his chest) and things go the same from there.
A major change occurs when Roshi advises him to search the world to see if he can further improve himself after being humbled by Gohan reminding him of his weakness, with Bulma going along with him. It's during these 3 years that Trunks is conceived (I'll say he's 2 by the time of the 22nd TB). Vegeta wins the Budokai again and isn't as fatigued as Goku was after doing so, meaning he manages to kill Tambourine and return to the group. Piccolo Daimao still births Cymbal and only chooses to interfere when Yajirobe kills him as well. Vegeta attakcs Daimao along with Roshi, though has his heart seemingly stopped and is saved by a nearby Yajirobe after Piccolo gets his wish. Things go the same from there until Kami's training, with Kami permitting Bulma to visit his realm or for Vegeta to go below as to raise Trunks. Vegeta's age and greater experience than canon Galu allows him to still beat Piccolo despite this. Obviously Chichi doesn't get her desire, so perhaps she ends up with Kuririn.
Without the Saiyans showing up, things are peaceful for a long time with Piccolo gradually becoming far less demonic and Vegeta both raising and training with Trunks. Regardless of him getting a heart virus or not though, Vegeta is easily killed with the other warriors by #17 and #18 years later. Trunks trains by himself and becomes a Super Saiyan eventually, though is aware he can't overcome the Artificial Humans and waits for his mother to create the time machine. This time, he acts far more secretive and gathers the Dragon Balls to find and kill Gero. Assuming his time travel shenanigans don't cause a Cell from another timeline to show up, he saves one timeline and uses the blueprints he finds from scouring Gero's lab (and eventually the secret lab) to shut down #17 and #18 in his timeline. Before that, he visits the Vegeta of timeline 2 and teaches him how to go Super Saiyan in case he needs it for a future threat. This overall proves fruitless however as Cell still kills Trunks in the future before going back in time (we'll say he just ends up in another timeline where #17 and #18 are still around like the Cell we know did) and Babidi's men kill the Dragon Team in timeline 2 10 years after Trunks' trip.