Scenario 1: Cell stays in his own timeline.
Cell and Trunks become allies and friends and live happily ever after. Meanwhile, back in the past Trunks returned from, Dabura yamchas everyone, so no DBS.
Scenario 2: Cell decides to travel to the past to fulfill his destiny.
Part 1
For simplicity sake, let's say he steals Trunkses machine without killing the latter and accidentally ends up in exactly the same time as in canon. He stays for 4 years in his larva form and awakens soon after the Z fighters have been defeated by the Androids.
Instead of going to Ginger to absorb people, he goes to the Kami lookout, where the Z fighters are gathered and Cell tells them his story. Kami senses his good intentions and the Z fighters decide to help him reach his 1st form's full potential so that he overpowers the Androids. Cell absorbs energy from Piccolo and possibly from other Z fighters, them being healed with senzus afterwards, until he reaches the power he had in canon after absorbing humans.
Cell convinces everyone not to destroy the underground lab, where his present self is growing.
A few days later, Goku recovers. Seeing how many guys stronger than Super Saiyans have appeared, he brings up the Spirit Room anyway. Vegeta and Trunks go to train inside.
The Androids appear at Kame House as in canon. Cell defends the house instead of Piccolo. Sixteen tells Seventeen not to fight Cell, for he's too powerful, but Seventeen doesn't listen. Seventeen loses, Sixteen steps in and Seventeen ends up absorbed the same way as in canon.
Semi Cell pushes Sixteen aside without using unnecessary violence and walks towards Eighteen, who threatens to self destruct. That declaration makes Cell start doubting whether the Present Androids are as evil as the future ones.
While Cell seems hesitant, Vegeta and Trunks show up. Vegeta tells Cell to quit dawdling and absorb Eighteen already, but Cell remains hesitant. Vegeta threatens to kill Cell unless he absorbs Eighteen. While Vegeta and Cell are arguing, Trunks decides to resolve the issue by destroying Eighteen himself.
Cell has an emotional breakdown. Trunks pats him on the shoulder and tells him it had to happen this way. Seeing that the sorrowful Cell is not responsive, he and his father decide to leave him alone and fly away, sparing Sixteen, who is in the same state of sorrow as Cell.
However, when the two Saiyans are about to go away, Sixteen ambushes Trunks and activates the bomb in himself in order to avenge the other two Androids. Sixteen selfdestructs, killing Vegeta and Trunks in the process. Cell regenerates, with a possible zenkai. He does not wish to stay on Earth anymore after all that has happened, so he leaves to explore the universe.
Trunks is later revived, with Vegeta staying dead, since Kami's Dragon Balls have already revived him once. Trunks returns to his timeline, where he becomes friends with that timeline's Cell.
Back in the main timeline, Gohan never meets Videl, since Satan is no world hero and, therefore, no Satan City is built. Nevertheless, Chichi convinces Goku and Gohan to participate in the Budokai, since, as she said in canon, her father had run out of money by that time. At this point, Gohan is his Cell Games self (with Goku alive, he hasn't completely neglected his training and trained just enough to maintain his ROSAT gains) and Goku is his canon self minus the SSJ3 form.
Goten wins the Kids Division, since Present Trunks hasn't trained with Vegeta and, therefore, is weaker than in canon.
Kaioshin and Kibito appear. Piccolo takes Vegeta's place in Babidi's spaceship and defeats Puipui on the first floor.
On the second floor, Goku fights Dabura. Kaioshin's like: 'Why is Babidi bringing out Dabura already? Does this mean he has an even stronger henchman in reserve?'
On the third floor, Babidi's secret trump card is revealed: Majin Cell! To be continued...