It could have worked in these ways...
1. Trunks is born to Vegeta and an unknown, female earthling.
In the original future, during the years the Androids were rampaging Earth, Gohan sensed a high power level and
thus met Trunks. Of course, he easily figured out he was Vegeta's son. The two of them train together as in canon.
Alternate Trunks goes to the past to warn the others. During his first travel (Mecha Freeza), nothing changes.
When he travels to the past for the second time, it is revealed he's Vegeta's son and Vegeta is not asked by anyone
about his personal life, but, for a moment, Vegeta gets an expression that shows he himself knows who the mother
is. Androids/Cell arc progress as we know.
7 years later, at 25th TB, an 8 year old boy that looks like the adult that came from the future shows up and introduces
himself as Vegeta's son. At first, he's mad with his father for not having stayed around to raise him. Eventually, their
relationship develops (when Trunks saves Vegeta from Fat Buu, thus showing he loves his father) and comes full circle
when Vegeta hugs Trunks (before knocking him unconscious to go and fight Fat Buu). The rest of the saga happens
as we know it, with Trunks staying around, fusing with Goten etc
After Buu Saga, Trunks keeps being a part of the group, but he does not stay at Bulma's house. He stays with his
mother, whose identity remains a mystery for the whole DBZ. I'm not against her identity revealed in DBS, though.
2. Trunks is born to Vegeta and an unknown, female alien.
During the years he served Freeza, on one of the planets he conquered for Freeza's sake, Vegeta raped a woman
and left without knowing that he'd left a child behind (so this Trunks is a few years older than in canon). After Freeza
died and his empire collapsed, Trunks somehow found out that his father settled on Earth and began a journey for
revenge against the one who raped his mother.
By the time he arrived, Vegeta was already dead and the Androids were rampaging the planet. Trunks decided to stay
to defend the earthlings and made friends with Gohan. They trained together as in canon. When Bulma made the time
machine, she had Trunks promise that, while in the past, he would not try to avenge his mother, for defeating Androids
is the priority.
Alternate Trunks slays Mecha Freeza and meets the gang as in canon. He shows a hostile attitude to Vegeta which no
one can explain. Their relationship develop later when they train together in ROSAT and comes full circle when Vegeta
gets mad over Cell killing Alternate Trunks.
After Alternate Trunks returns to his timeline and kills Androids/Cell, he returns to his home planet where he tells his
mother what happened and she finally forgives Vegeta as well.
Somewhere during the 7 years between Cell/Buu arcs, Present Trunks arrives on Earth for revenge, as he did in the
original timeline. A mini saga follows with Present Trunks being overpowered by Vegeta. He decides to stay on Earth
until he finds a way to take his revenge, but he is still friendly to the other Z fighters.
At Buu Saga, Present Trunks is part of the events. Perhaps he is majinized and takes Goku's place in the canon duel
that awakened Buu. For fusion saga to work, Gohan takes Goten's place and fuses with Present Trunks and Goten is
the one taken to Kais' planet to draw the sword.
In DBS, Zamasu/Black Goku eventually reach Alternate Trunkses home planet. Alternate Trunks had a capsule with
a time machine and goes to the past to fetch help. The battles against Zamasu/Black Goku take place on Alternate
Trunkses home planet, but, in the end, Zeno erases the universe anyway.
