Vegeta got his wish on Namek?

[GSM123]

Let's say Guru's lifespan wasn't cut short by depression and Vegeta got his wish. How differently would things go?

 

[Captain Cadaver]

Assuming immortality is the same regardless of the Dragon Balls (ie. Garlic Jr/Zamasu tier), Vegeta essentially has complete regeneration and, thus, is able to abuse what Zenkais he has left. Given how it's implied Vegeta ran his Zenkais out around the end of the Freeza Arc or sometime prior to the Boo Arc, he still wouldn't end up becoming strong enough to compete with True form Freeza's higher percentages. Though undying, Vegeta is overpowered by Freeza after he quickly goes into his true form and the rest of the group are killed before Goku gets the chance to arrive. When Goku arrives and sees his son, best friend and Piccolo are dead, he becomes a Super Saiyan. Freeza attempts to destroy the planet, though Vegeta gets in the way and takes the attack before regenerating, with Goku deciding to kill Freeza to avoid that happening again. Vegeta decides to return to Earth with Goku and Bulma. With the severe losses, Chichi divorces Goku and the general tone of the arc's ending is a depressing one, with only Goku and Vegeta as the active fighters remaining. The story then takes one of two routes:

1. The Standard Route

In the year since Bulma has been forced to accept Yamcha's passing, Vegeta staying with her has caused them to grow closer at a quicker rate, though it has yet to become sexual as Bulma would still need a significant amount of time to grieve. King Cold is stopped when coming to Earth. Goku either dies from the Heart Virus or is killed by #17/18 and when faced with the two Artificial Humans, Vegeta plays dead and prevents his regeneration until the two leave him alone. When they're gone, he heads to Capsule Corp. and tells Bulma to take Trunks and leave with he and Vegeta on a ship. They then set up home on a new planet where Vegeta trains to achieve Super Saiyan and become strong enough to beat #17 and #18. With the necessity to preserve and expand the Saiyan race, Bra would also be conceived far earlier. Training Trunks and Bra over the years, he eventually feels himself ready to do so. Let's say it's at the point in the timeline where the Boo Arc would normally occur. He takes the ship by himself in case the twins attack his family and faces the pair. With the power he's gained through training, he destroys the two and brings his family back to Earth, having come to accept this will be the new home of the Saiyans and that he must live as a Saiyan and human.
A few years later, Shin and Kibito come to Earth and tell Vegeta about Babidi. Vegeta chooses to confront Babidi outright and is spat upon by Dabura, being completely turned to stone. His immortality useless now, Vegeta's role in this battle is over. Shin takes Trunks and Bra to be trained on the Kaioshinkai, as well as Bulma for her protection. For many years, Dabura terrorises Earth and enslaves humanity to be a constant source of his energy. By the time Trunks and Bra have become strong enough to pull out the Z-Sword, Shin would've become anxious and say there's no choice but to go now. Unfortunately, as suggested by Super, Dabura is far stronger than his Boo Arc self and easily kills the two Saiyans and Shin. Babidi eventually revives Boo, who kills Dabura. In doing so, Vegeta comes back to life and fights Boo when given little alternative. When interested that Vegeta has similar regeneration to himself, Boo ends up turning Vegeta into chocolate and eating him. Vegeta reforms inside Boo and wanders around, eventually finding the pods. I imagine that with the different structure of Fat Boo's body, Dai and South Kaioshin would be the ones present. Vegeta removes both and Pure Boo is formed. The Kaioshin quickly wake up and succeed in destroying Boo. They then return to the Kaioshinkai, where Vegeta reunites with Bulma.
With all that's happened with the death of Beerus, Super as we know it doesn't occur. When Moro escapes, Dai Kaioshin would attempt to seal him. I believe it would work this time though as in being removed directly from Fat Boo, I don't believe his God Ki would've been given to Boo. Even if not, Vegeta would learn the Mafuba from Roshi and do the job.

2. The Manly GAINZ Route

Seeing the situation, Roshi decides to train again and quickly becomes as strong as his Super self, saving the Earth from all threats with his Pseudo-Ultra Instinct Manly GAINZ.

 

[withheldforprivacy]

Freeza transforms into his 2nd form and mortally wounds Vegeta, who survives because of his immortality. Vegeta gets the same zenkai as in canon and kills Freeza. Then he waits for Goku to be healed. When he sees how strong Kakarrot has become, he decides to follow the gang to Earth instead of trying to take control of Freeza's empire. Eventually, everyone is revived.
Goku dies of the heart virus and, 3 years later, the Androids kill everyone except Vegeta, who eventually finds a way to defeat them. He and Trunks are the only Z fighters left.
Babidi/Dabura are easily disposed of.
Beerus probably never wakes up. If he does and Vegeta somehow unlocks SSJ Blue by training with Whiss, no FNF, since Cold is alive and the empire is not in chaos. Vegeta solos U6, unless he goes SSJ Blue against Cabba, so he loses against Hit because of stamina issues. If Zamasu gets jealous and takes Vegeta's body, the Zero Mortal Plan succeeds, since there is no Future Trunks.