Goku would probably use Shunkan Ido to get to Earth when sensing his friends were in danger and try and talk Trunks out of it. Trunks would likely come to a compromise that he'd leave Earth alone and attempt to reconstruct Freeza's Empire into a benevolent monarchy of his own in order to maintain peace across the Galaxy, since I can't see him being as big a hypocrite as to inflict the kind of terror #17 and #18 did to his species on others. Cold is still kept alive as someone is needed who knows how the empire is ran, though Trunks watches him vigilantly. Before departing from Earth, Trunks informs Goku about the Artificial Humans and other pieces of information.
The next 3 years, he spends ruling the galaxy and making various reforms to turn the Freeza Force into a more peaceful and stable rulership, now dubbed something along the lines of "The Temporal Dynasty". Because of this, he doesn't get as strong as his canon self did, though that doesn't matter when his role in the plot would remain unchanged, plus he'd at least make considerable gains by sparring with Cold every now and then. He and Cold would depart to face the Artificial Humans when the time came, likely arriving sooner due to the lack of Time Machine factors would make it easier to deduce how long he'd take to arrive on Earth. We then have two routes this can take.
1. Let's say he arrives around the time Goku's heart virus kicks in. In realising these unknown enemies are different Artificial Humans, he goes after Gero while Cold would jump in to face #19 only for Vegeta to arrive and take over that battle. Trunks would still likely have grown strong enough to overpower Gero and would be given a head's up by Yamcha about energy absorption. Vegeta kills #19 and Trunks backs #20 into a corner. Gero's plans are foiled, #17 and #18 are destroyed and even if Piccolo and Kami don't fuse, Cell's absorbing spree would lead to him being cornered in Gingertown and beaten by the SSJs. Trunks then has thoughts about returning to his own timeline, but after judging how much better he's made the galaxy as a whole in this world, he chooses to stay as a monarch. Without going in the Rosat, however, everyone is killed by Dabura 7 years later, assuming Goten and Present Trunks aren't as plot boosted as their canon selves.
2. If he arrives around a similar time to his canon self, the arc remains virtually the same except with Cold being a tag-along until the Rosat. Cold wouldn't pass up the chance to enter the Rosat in an attempt to reclaim his former glory through new power, likely going in during Trunks' 2nd trip. Trunks would make a note that he would kill Cold if he got too strong, since his ambitions likely wouldn't have changed after this time. Whether or not he could make major gains is questionable, seeing as how he seemed pretty old (and doesn't know da wae of Manly GAINZ), so I'd say at best he'd be slightly below Vegeta/Trunks. Only thing that changes between then and Cell's death is Cell making 8 Cell Juniors. Trunks still stays in this timeline.
As for the Boo Arc, Shin would come to Trunks to fill him in on the details before heading to Earth due to Trunks' influence. He has his top soldiers, Tagoma and Shisami, observe Earth from space while he and Cold rejoin their allies at the Budokai. Whichever two jobbers they replace in it doesn't matter, since the overall outcome is the same. When it comes to Babidi's ship, it would likely be Trunks who joins Goku and Vegeta instead of Gohan, since I can imagine Trunks as keeping up his training and having surpassed his rusty friend by then. He defeats Dabura and the Boo crisis is averted. As for the plot of Super, SSG would still happen even without Boo, just a little later and not under hostile circumstances, as well as far easier with Future Trunks available. FnF never happens and the U6 Arc is relatively the same.
The Future Trunks Arc is far different for obvious reasons. Not able to defend his timeline, Trunks' world has it's inhabitants killed or made as constant enery resources by Babidi and Dabura until Boo's revival. Whether or not this happens doesn't matter as Zamasu and Black would kill them and complete the Zero Mortal Plan for that timeline before moving on to another. Without anyone to inform anyone, the amount of timelines and universes affected by Zamasu would be incalculable. Either Zamasu succeeds in wiping out all mortals or Zen-Oh/Daishinkan erase him before he can do much damage. In the case of the latter and the ToP, Trunks would replace Freeza (and would likely have achieved Blue through training with Galu and Daddy Veg), overall being a far greater asset regardless of whether this is the anime or manga events due to not messing around like Freeza did.
But would King Cold tolerate a Saiyan bossing him without him trying to assassinate by covert means(maybe poisoning?)
