Since Vegeta steps up a lot earlier, Gohan and Kuririn get killed and Piccolo gets beaten to the point he can't retaliate before Goku shows up. The fight goes the same until Vegeta goes Oozaru, at which point Goku has no chance of survival. Even if Yajirobe cuts off Vegeta's tail, Vegeta still wins the battle easily before interrogating Piccolo. Upon finding out the Dragon Balls won't be active for another year, he kills Piccolo and heads to Planet Freeza #79 to heal, before heading to Namek.
With his wounds being nowhere near as severe and the Earthlings not being there to distract Freeza (even if Bulma goes alone, she'd be killed or captured by Freeza's scouts), Vegeta would have to hide from Cui, Dodoria or Zarbon when getting the Dragon Balls. At best, he may be able to get his 30k Zenkai and kill Cui, Dodoria and Zarbon as well as get the Dragon Balls, but without any Namekian to help him and one of the only two left being capable and likely of killing him, Vegeta can't win. Freeza kills Vegeta himself if Nail doesn't get to him first and after realising there's no negotiating with the Namekians, decides to destroy the planet. He continues his reign unless running into Babidi or Beerus wakes up. or if he decides to visit Earth in a few years' time.
As for back on Earth, Gero would still create the Artificial Humans, though probably change his plan to purely world domination rather than revenge. Regardless, #17 and #18 still wouldn't follow him and the Earth becomes akin to what Trunks' timeline would be if nobody could give them the slightest resistance. As long as Freeza doesn't show up immediately after destroying Namek, the twins are the dominant force on Earth until Cell appears and absorbs them. After that, he'd probably head out into space to find more opponents, possibly finding and killing the likes of Freeza, Cold, Dabura, etc. He'd remain unchallenged unless Beerus chooses to wake up (or he runs into Canon
).