I've been pondering this for a while - somewhere between theorising and drafting fanfic that'll probably never be written, but I assume the original plan was a lot simpler. Warp the tower to the First, then summon the WoL and play the part of the "mysterious wizard from another world" who has always lived there in his crystal tower. (Oh, there's a matching one where you came from? How interesting.) Guide the WoL to defeat the Lightwardens, do his betrayal-and-warp-to-the-void act, his death undoes the summoning and returns the WoL to their own world. The worlds are saved, the villainous Exarch is undone by his own actions, and the WoL never learns his true identity.
Of course it's nowhere near that simple, and as soon as G'raha lands in the First and the people come to the tower seeking aid, he couldn't possibly turn them away. So the years pass as he nervously watches events in the Source play out in slow motion and wonders how long it will be until he reaches the right point in time...
And then he goes and accidentally pulls the Scions to the First, so now instead of being alone, the WoL is going to have friends who will undoubtably try to interfere when the Exarch does his villain act. And thus Urianger gets roped in as secret-keeper.
Could the plan have worked with the WoL alone, and G'raha only recently arrived to the First himself? Perhaps he wasn't expecting such a thriving world, or simply that it would be much closer to the chaos immediately following the halted Flood.
But as for the timing, assuming that he thought he only needed a short time to prepare, then he arrived just a few months prior to the necessary point as calculated by the Source's flow of time. If the relative speed fluctuates between the two, he might have to act quickly once he got there, so he'd want some leeway with it... and instead that leeway turned out to be decades.
(Imagine what that would have been like... watching things play out in slow motion all those years, never sure if the flow of time might suddenly speed up and he'd miss the critical moment.)
What happened to Emet in the other timeline, you mean?
He certainly would have been active - remember he had to come back into action because of Lahabrea's failure back in Heavensward. Elidibus would have needed to call on him in either case, even if you're of the theory that the timelines diverged as soon as G'raha and the tower arrived in the First.
The only way I could see him being "involved in G'raha's plan" would involve him finding out about how the time travel mechanism works, and I think that would be too catastrophic on a larger scale in other ways.
Though I was toying with the idea that if the split in the timelines didn't happen until right at the end of Shadowbringers - hinging on whether we defeated Hades or not - then in the other timeline the WoL's death might just bring G'raha to the point that he spills the whole thing to Emet in anger and "do you realise what you've just done?" and... I'm not sure where things would go from there, but certainly in the short term it would be in Emet's best interests to work together for at least long enough to get G'raha out of the First alive, and possibly make sure that the Ironworks and younger-G'raha do manage to pull off their time-travel venture so the other timeline can be saved, even if this one is ruined and the Ascians' plan can't be fulfilled either.
Of course, as much as I would love to be idealistic and think that Emet would have a change of heart... I greatly doubt it. Most likely he'll play along until he can swipe the time machine and go back to Amaurot to halt the Final Days. And that probably isn't the outcome we're looking for.
Most idealistic thought though: using the tower to return to the point in time that G'raha left from, for a reunion with the friends he had to leave behind there, and just maybe they can find a resolution for healing the world from Black Rose. Perhaps if Hades made an early escape and G'raha made it back to the tower alone...?