Let's add a friendly image so that people don't see any spoilers by accident.
So, the way I see it the Emperor probably tailed Shadowheart's team who liberated the Astral Prism. Maybe he even got the Nautiloid and some cultists and Mindflayers from the Elder Brain so that he could extract Shadowheart's team. Orin left her sibling on there for good measure. Mindflayers doing Mindflayer things they abducted Shadowheart.
But as soon as the Prism was close enough the Elder Brain lost its grip on the Emperor. Probably just him because he already was able to leave an Elder Brain once. Seeing the opportunity he killed the Mindflayers near him and took control of the Nautiloid.
Being the practical Mindflayer he probably sought to build an army of tadpoled people with super duper Illithid powers to fight the Absolute. He went to Baldur's Gate to find more people and thus snatched up Tav and the others.
The Githyanki were in close pursuit and disrupted him so he had to flee. When the Nautiloid arrived on Avernus he hid inside the prism.
I'm not quite sure where Lae'zel fits into all this. She probably went aboard the Nautiloid to get her Mindflayer head but was captured instead. She didn't know anything about the Prism because it was way above her paygrade.
What do you think happened? Or is there an official version of the events that lead up to the game?
It’s explicitly stated by the Netherbrain in act 3 that they allowed the Emperor to escape their thrall, anticipating all of the events leading to him discovering Orpheus and your adventuring party - as well as the the chosen of the Dead 3 discovering the Crown of Karsus and the Ilithid oubliette under Moonrise Towers. It is implied that the whole idea for the creation of the Cult of the Absolute and planned power grab by the Dead 3 was also anticipated/orchestrated by the brain from the beginning.
That is stated, but I'm not sure I'd consider the Netherbrain a reliable narrator. I feel like it was an attempt to intimidate the party. The sheer amount of secret knowledge and unlikely events that come together to begin freeing the brain at the end of A1 seems beyond the ability of an elder brain, which is all it would've been when it formed the plan.
The whole idea that it would use use an artifact that had only been seen once, very briefly, with no measurement or understanding of its true capabilities, to evolve itself into a being powerful enough to overcome the control it was submitting to as part of its plan... I know elder brains are scary smart, but this seems a ludicrous proposition.
Consider: what part of its plan could it not have achieved simply by tadpoling Gortash and anyone else to begin with?
Now it could be that what the brain says is canonically true and this is just weaker writing, but until I see confirmation I think the brain was trying to demoralize the party into thinking they'd always been dancing to the brain's tune the whole time and everything they had done had been in service to its plot the whole time, when the truth is it has been (very capably) trying to regain the upper hand since Ketheric's death.