This is tempting me to write up a fanfic...
The Wizards, initially known as the Istari or Heren Istarion (Order of Wizards), were five Maiar spirits sent to Middle-earth while embodied as old Men to aid the Free Peoples against the threat of Sauron. In the Undying Lands around TA 1000, Manwë (leader of the Ainur, and King of the Valar and...
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All five appear sometime after Aegon’s Conquest, occasionally advising the Targaryen Monarchs on the best course of action
I believe Alatar and Palando would head up north, meet with the Starks and then venture beyond the wall, or perhaps go East and then to Dorn...They'd almost certainly accompany Saruman to Ashai by the Shadow. Radagast the Brown would certainly take up residence either in the Reach or in the North. His knowledge of beasts would certainly be prized by the citadel and I could see many an Arch Maester sending his novices to go spend a summer or two with the old wizard.
Saruman and Gandalf would spend some years in the court of Aegon the Dragon, likely spending a lot of years going over Valryian lore with Aegon and very likely rediscovering Valryian steel and how to forge it as he was a Maiar of Aule and mad science and engineering is his first and greatest passion.
Gandalf I think would likely be the one who spends the most time with their Dragons aside from Radagast with King Aegon surprised that Belearion seems to "enjoy" conversing with the gray pilgrim. Over the years, their agelessness and seeming immortality would be a source of great concern and resentment. My guess is Meagor the cruel likely tries to kill one of the Istari and pays dearly for the attempt.
I think Saruman would likely go to the Citadel and advance their knowledge, hang out with all the sorcery bros and then fuck off to the East to learn of their magics, while Gandalf would study along as well, it wouldn't be his chief concern and he would probably end up sporadically tutoring Prince Jeaharys.
By this point, Saruman would likely find himself on the small council and Gandalf, would be seen often as well. One would likely become "Master of Lore" or something...A hand's position, would be unthinkable as people would chafe at the idea of his timeless mage taking such power.
Moving around the lands of the Lords to their annoyance
They wouldn't start wandering in earnest until likely after the death of Jeaharys, whereupon Saruman would likely depart the council to "venture east to find my blue brothers". He would very likely not be seen again until the reign of Daeron the good...And that's even assuming the Dance of the Dragons happens at all with Gandalf around...He may just fuck off and say nothing or he might rage at the lords for disregarding an oath, he might curse both sides...with an ominous warning.
Saruman would return, likely a generation later and set to "putting things to order" which at first the real would heartily agree with..but.
Guys like Saruman occasionally blessing the Citadel with his knowledge and guys like Gandalf occasionally showing the smallfolk things called “fireworks”
He wouldn't occasionally bless them, Saruman would outright take the Citedel over or if he's still being subtle..turn it into an order that is almost entirely devoted to him when he makes his return.
Being hit by staffs may occur
Mind raping, voice fucking, being set on fire, being drone struck by blackpowder delivered by an unusually large Raven..etc etc./
How do the leaders of Westeros and Essos respond longterm?
They ignore them at first, seeing them as more eccentricity from Aegon..then they get humbled and listen..then they chafe, then they get resentful...then enough generations pass that they get integrated into Westeros.
It helps that the two most powerful might appear once every thirty or so years, until Saruman decides to stay around for keeps.
I’d bet Gandalf would do something to prevent the Blackfyre Rebellion
Why would he? Gandalf didn't run around preventing civil wars in Middle Earth and the Istari spent about a thousand years roaming the world of men. He helped where he was needed, he marshaled the forces he believed would be best suited to contest what he felt was the rising power of Sauron. Sometimes that meant destabilizing the established order...to suit his needs.
It's entirely possible the Istari would side with Deamon Blackfyre on the condition that the Targaryens be kept around as a cadet branch and be given lands to rule, even if that means conquering Essos or the Stepstones or someshit. Hoping to gird the realm against the forces of the snow fairies or the Others or someshit.
Hell, it's debatable that there'd even be a Blackfyre Rebellion due to the fact that there may not even have been a dance.
Something to note- Olorin's point of contention against both Sauron and Saruman's vision of imposed order was that it was needlessly destructive..and indifference to the common man....Not that they wanted to impose order. Tolkien himself even notes this and says Gandalf would be worse than either of the two if he ever went batfuck people would love him for his tyranny.
There's a very good chance....In this setting....Gandalf is just as zealous as Saruman..only his efforts would be in establishing more of a Westerosi "middle class" than anything else...