What happens? What House does he side with? Does he decide to go for the Iron Throne himself, knowing he's the most powerful being in Westeros? Or does he bide his time as a counselor to the humans? Would he survive the politics or is he desposed of quickly?
What are your theories as to what the Grey Pilgrim would do.
Hey Edmond. To answer this question meaningfully, we have to re-examine what Illuvatar sent Gandalf to do on Middle Earth in the first place. Now, perhaps you could help me since your the LotR expert, but my understanding is that his job was the curtail the influence of Sauron and, my extension, Morgoth. On a technical level, he is supposed to allow human failings to proceed on their own terms, since humans are the creation of Illuvatar and thus them screwing up was just things playing out how they were meant to. Morgoth imposing his own designs, that was the issue.
So, in order to necessitate Gandalf's existance in Westeros, that atleast implies that Morgoth has either touched that realm as well, or that the supernatural forces of that world are a sort Morgoth equivalent. It's hard to tell who that is as the divinities of Westeros are somewhat vague, so for the sake of argument, lets assume it's the white walkers. That means it's his job to stop them.
He essentially has the same job that the Nights watch then, obstensively, which is to convince the people in power to act against the White Walkers. So at that point, it's mostly a matter of choice between trying to convince King Robert himself or else a House that is powerful enough and power hungry enough to listen to this. I actually think his job would be rather easier in the world of Westeros in this regard because he would have a monopoly on magic until Daenerys' dragons are reborn. All he would really need to do is flash some of his pizzaz and any house he chose would be clamoring for his employment as a Maester.
What I actually find more interesting is what would happen to his character. In Tolkien's world, Gandalf has friends in the form of hobbits and various people, and his enemies are easily idealogically defined. He had to deal with the betrayal of Saruman and the downfall of Denethor, but on the whole, he has formed genuine bonds of friendship with hobbits, with Aragorn, with the elves and the dwarves. With Middle Earth, Gandalf knows who he's fighting for and who he's fighting against and why.
In Westeros, we still don't actually know what the White Walkers even want to get by killing humanity (well, maybe the show watchers do, but not in the books). They're just zombie monsters, horrifying but unknowable. But because they've been relegated to bedtime stories, the people ofWesteros have largely turned their competing interests against each other. The politics of Middle Earth weren't happiness and sunshine, but they certainly weren't as cut-throat and malicious as they are in Westeros. So, the real question here is, what would Gandalf have to do in order to ingratiate himself to people like these? Sure, the Houses would be clamoring on top of each other to get his employment, but once he chose a house to aid, how would he then make alliances in the others? Some houses, like the Starks and the Tyrells are more chill and could allow genuine companionship to grow, but how would he make friends with the Lanisters?
He would either have to use his power to influence people more than he ever had to in Middle Earth, where he was able to serve as an advisor and aid to kings instead of outright commanding them or else he'd have to play the Game of Thrones and start pulling genuinely underhanded tactics. And the real question would become how does the only Wizard in the world keep his soul when everyone around him has long since lost it.
In fact, I think the reason he got along with Hobbits so well is because they were genuinely seperate from the political BS that he had to deal with when he's on Valar business or having to deal with Gondor and Rohan and the Elven-Dwarf relations. When he wanted to kick back and just chill with his friends, he went to the shire and talked with the fat little people whose biggest political tensions were that Otho and Lobelia were kind of assholes. One of the largest appeals of the Shire is that it's utterly hidden away from the rest of the world. The Hobbits grounded, relaxed, and humanized him. Who would play that role in Westeros, where the politics were far pettier, far more ruthless, and utterly inescapable?
And if the answer is no one, then how does Gandalf not end up turning into another Saruman? Especially since diminishment doesn't apply. Tolkien made sure to craft his metaphysics so that evil is consistently punished by the character becoming lesser when straying from Illuvatar's intentions. Morgoth became weaker, Sauron became a shadow of himself, Smeagol became Gollum, and by the end of the series, Saruman was reduced from a grander standing than Gandalf himself to being beaten and driven out by Hobbits he held in contempt.
In Westeros, this doesn't happen, or it doesn't happen BECAUSE you make an evil choice. If Gandalf did become corrupt and selfish, and took power....that doesn't come with any metaphysical cost. He might have set backs or be overthrown via machinations of rival factions, but in all likelihood, it would just mean that what it says on the box: That Gandalf takes power and maybe becomes a worse person, but his standing and power would not likely be diminished like Saruman's was. Saruman's ambitions would instead become his reality.