As
Hobbits are not an aggresive species by nature, I dont think that
Bilbo would have killed
Gollum after losing the
Riddle game, except maybe through self defence.
This leaves two scenarios:
1. Bilbo gets captured by the
Goblins, and tortured as Beleg so rightly stated.
2. Bilbo miraculously finds the way out, and heads off as normal.
The Ring goes back to
the Shire, and is handed over to
Frodo. The Ring is taken to
Rivendell, through
Moria, much as it did before, and all the way to Mount Doom, without much struggle (would make TTT a bit boring to read). Frodo has no enemy to fight with at the top of Mount Doom, so Frodo keeps the Ring, and he turns bitter and twisted the same as Gollum did.
Frodo goes and hides out in a cave, after being expelled from
Hobbiton, living on a diet of fish, with his "Preciouss" and having his life prolonged by the Ring.
One day, he loses the ring and a curious
hobbit finds the ring whilst trying to escape some goblins. the hobbit wins a game of
riddle with the weird looking Bilbo, and takes the ring back to the Shire.
After a number of years back in the Shire, the hobbit, hands the ring to his heir, who takes on the task of taking the ring to Mount Doom to be destroyed.
He meets up with some people in Rivendell, they walk off and start their adventure.
Bilbo gets captured by
Sauron, who then dispatches
the nine, and the whole story starts off again.
You cant change fate I guess!