Vegeta takes a gazely stare at Goku defeating Freeza with SSJ, and looks pretty upset. He sees that he himself, the prince of the Saiyajins, is not only surpassed in the base form but also does not possess a transformation that increases the power of the user by several times. He promises that he will surpass Goku and that he will once again be the Saiyajin # 1, the latter which, much like in the main timeline, takes a look at Vegeta and says "whatever."
Vegeta then begins his training journey, however not as rigid as he did in the original timeline due to the lack of upcoming threat (or that he thought), which certainly increased his will to get stronger. Vegeta does not ask Dr. Briefs to build the gravity machine and prefers to leave on a training journey just like our dear Namekuseijin, also meditating on solitude.
And then, he gets a heavy blow: "Goku is dead!" Vegeta is very shaken and kind of upset by the news (which occurred only a few months after he begun his training journey) and, without a rival to overcome/surpass (Goku), without a need to survive (androids) and being the strongest on Earth, decreases a lot the intensity of his training that, as already said, was never the same as the one performed in the timeline we all know. He ends up by not trying and struggling in his own misery so hard, does not experience a huge anger towards his lack of success and, as a consequence, does not become the legendary Super Saiyajin. Much like in the time line we know of, he has sex with Bulma and has/gains/makes his first (and only) son, who is not as powerful as Chibi Trunks from the main timeline due to Vegeta's lack of Super Saiyajins sperm.
Then the terrible androids arrive. Vegeta (as well as the other Z warriors) decides to face them and suffers an embarrassing defeat. In the midst of the fight, he realizes that Piccolo (who also kept his training, but not with the same intensity as the main timeline for the same reason as Vegeta), has gotten stronger than him, due to his increased potential granted by the "Namekian fusion" with Nail, granting Piccolo potential on par with the SSJs ... he then has a sad death knowing that he ended up dying being just the second strongest and, completely poweless before the killer machines. Poor future Vegeta!