Assuming that Goku died in episode 1 of DB....

[Rocketman]

Bulma is a horrible person, so after shooting Goku in the head she plunders his house and takes his Dragonball. She then gets eaten by a pteranodon.

Turtle dies miles from the ocean.

Oolong, Yamcha and Carrot Boss continue terrorizing their respective areas.

Pliaf is screwed. He won't be able to get the Dragonballs from Ox King or Roshi.

Krillin still arrives, goes through training and places well in the Budokai. Roshi may throw the championship fight to Nam.

If/when ChiChi ever gets to Roshi's isle, he'll put out the fire.

Lunch is shot dead in the canyon.

The Red Ribbon Army easily kills Pilaf and takes his DB, as well as the three that've been laying around since Bulma died. Oolong goes down too, but then they run into Roshi and Ox King.

It seems like a setup with Roshi, Ox King, Krillin and ChiChi on one side and the RRA on the other, with General Blue their heavy hitter.

Without Goku's constant meddling, it takes Red longer to hire Tao. If he does, then it depends on if you think Roshi could beat Tao.

Either way, Red's wish is to get taller, not rule the world, so even if Tao wins there's no guarantee the RRA manages to conquer the world.

Then Raditz lands and everybody dies. If he somehow finds out about the Dragonballs (maybe somebody tells him in hopes of mercy?), he tells the other two and they make their way to Earth.

With nobody left who can fight them, the Saiyans get immortality (assuming Kami doesn't intervene and destroy the DBs). They destroy the Earth, then head for Namek, following Vegeta's remembering of rumors. Freeza also heads there to beat them to it.

What happens next depends on how immortality works. If they gain a healing factor like Wolverine or something, the Saiyans are unstoppable and zenkai-hax their way through anything Freeza throws at them, up to and including Freeza himself.

If they still have to heal at a regular pace, that takes long enough for Freeza to get the DBs. But, he can't speak Namek, so he's shit out of luck.

Gero died with the Earth, so no Androids. Buu is odd. Even if Babidi finds the egg he still has to drain ki to activate it, and it's very doubtful there's anybody near SS2 Gohan in this universe.

Immortality means the Saiyans are an infinite source of ki, but also means they may grow strong enough to kill Dabura before Buu is revived.

[Attitudefan]

She then gets eaten by a pteranodon.

This is technically the first bad guy in Dragonbal and everybody forgot about him, including me. So Bulma would be dead as they did encounter each other and not having Goku around would mean that's the end of Bulma.
Rocketman's explanation fits it best.

I would like to say that if the planet is destroyed, Buu's egg thing would have probably gone with Earth, gone to another dimension

[GS7X7]

I don't know, given how powerful Buu is I don't think it's that easy to destroy the egg thing.

Besides, suck up enough energy from a few million or billion people and you can still power Buu's release.

Also, an immortal super-strong Saiyan wouldn't necessarily be enough to stop him.... Buu easily tricked Gohan and made him into his bitch and absorbed all his power, if a super-strong Saiyan came along he could just pull that trick on them regardless of their immortality.

(speaking of which, we seem to have forgotten Garlic Jr. and the movie 2 scientist and how they'd change things in a Goku-less world)