Tyrion, Brienne, and Jaime
How the story was conceived before Martin starting writing it and what it evolved into is vastly different. This is no surprise. Nothing in the story has changed more than the story lines of Tyrion and Jaime. The article on Broken Plots goes into these changes more extensively, but due to how dramatic the alterations are to the story arcs of the Lannister brothers requires a separate article on them. And I can't talk about Jaime without discussing Brienne.
Martin's submission to his publisher gives us a starting point, though not much detail. What I am going to go into is what almost certainly has played a role in Martin's writing. Martin has famously described how the image of Bran witnessing an execution, which formed the core of Bran's first chapter in the story is what inspired him to write ASOIAF.
This may very well be what started it, it is not the only thing he imagined before he started writing. These additional aspects, we might describe them as signposts, that help direct his rather un-directional writing.
Exactly which ones they are can only be guessed, but we can infer at least some of them. Dany's visions in the House of the Undying is one. Quaithe's prophecies to her is another. The Kingsguard, the Night's Watch Oath, Aegon and his sisters, Valar Margulis/Valar Dohiris, the Red Wedding, and the Valonqar Prophecy.
Our focus here is the Valonqar Prophecy.
By Martin's own admission Jaime as originally going to be the Big Bad. When I realized that the Valonqar Prophecy referred to one person and not two it changed the entire meaning of this prophecy. The "younger, more beautiful" person was not Daenerys, Margaery, Sansa, or even Arya. It was Jaime.
And yes, that does mean that it was going to be Jaime who would be responsible for the deaths of their, his and Cersei's, children. This suggests that it would have been Jaime and not Olenna who would have conspired with Littlefinger. It also suggests that Tommen and Myrcella were going to be 'the princes in the Tower'. I'll explain below.
By this version of Jaime, Cersei would have been the stereotype female sidekick. Boris and Natasha you might say. Her devotion to Jaime would have been total. A complete opposite of what we actually get. Until that is she discovered that it was he who murdered their children. Then she would turn on him, killing him with the Catspaw Dagger. He in turn would get his hands around her throat, thus fulfilling the prophecy of them dying together.
It doesn't take much mental strain to guess that this story line did not last to the end of the first book. As for why it goes back to the 'ultimate villain' trope I go into in my Harrenhal article. Just as with Littlefinger, this construct is a little too close to the mustache twirling villain. Martin hates these clichés.
By the time he had finished A Game of Thrones Martin was already shifting Jaime's story-arc to one of redemption. To understand how we first have to go on a tangent.
Brienne was not one of the characters Martin first imagined when he began the story. This is hardly surprising as a great many of the characters have been created as he went along: Ramsay, Hizdhar, Gerold Dayne, and so-on.
Why she was created takes a little more figuring. It likely has to do with Renly. Or more specifically, with how Renly appears in the story.
It is easy when watching the show, and even to an extent reading the books, to assume that Renly was just a puppet for the Tyrells. A male version of Sansa. But that isn't how he was conceived. Renly was originally meant to be a major character. If the War of the Roses analogy holds true he was going to be George of Clarence.
(The Ancient Origins website lists George as Theon. But this doesn't hold up as there is never any question of who Theon's family is. George betrayed his family out of ambition. Theon merely returned to his family.)
That means he was going to switch sides once the Tyrells turned on him. We don't have to concern ourselves over the mechanics of how this was going to work because obviously Martin himself couldn't figure it out. What matters is that Brienne was created specifically to give Renly agency. In other words, to free him from the Tyrells and possibly to act as a foil for Margaery.
This purpose did not last past the chapter in which she is introduced. By the time Brienne enters the story I think Martin had already decided to cut Renly out of it. From the beginning Martin has been struggling, not always successfully, to keep his story under control. Cutting Renly's greater story-arc is one of them. As mentioned above, trying to make it work in the context of the greater narrative was just not worth it. Also, it gave Martin a chance to add more magical elements, which are rather lacking in the story up to this point.
I have seen comparisons made regarding the characters in ASOIAF and their War of the Roses counterparts. But I have only seen them regarding a few specific characters: Robb as Edward IV, Daenerys as Henry Tudor, Ned as Richard of York, ect. But there are a few I haven't seen. Stannis was going to be Richard III. As mentioned Renly was George of Clarence. Originally Tommen and Myrcella were going to play the roles of the Princes in the Tower. If it feels like they don't fill those roles its because Martin changed the story to omit them.
We should list it like this:
Edward IV Robb Stark
Richard of York Eddard Stark
Margaret of Anjou Cersei Lannister
Henry VI Robert Baratheon
Edward of Westminster Joffrey Baratheon/Waters
(I don't know what his bastard name would be.)
Elizabeth Woodville Jeyne Westerling/Talisa
Richard III Stannis Baratheon
Ancient Origins lists Roose Bolton as Richard Neville, the Kingmaker. This is totally incorrect. Although Roose is involved in killing Robb, he can hardly be described as a kingmaker. Instead Warwick is split into two different characters:
Richard Neville Jaime Lannister/Petyr Baelish
Baelish represents his double-dealing side and Jaime is his warrior side.
Warwick was not the only person who was replaced by two characters. Henry Tudor also went into two characters: Daenerys Targaryen and Jon Snow.
Ancient Origins also mentions Bran and Rickon as the princes, but this is also inaccurate. Nor was it in the original telling. The death of the princes was supposed to grease the skid for whoever seized the throne. Neither Bran nor Rickon ever presented any obstacle for such. That is why it is actually Cersei's youngest who are the 'princes'.
Edward V & Richard of Shrewsbury Tommen & Myrcella Baratheon/Waters
So on choosing to end Renly's participation Martin finds another purpose for Brienne, which incidentally elevates her character from being just a sidekick and into a major role. Brienne now became the vehicle by which Jaime discovers redemption. Incidentally, she also became a foil for Cersei. Yes, there is a definite 'Beauty and the Beast' vibe here. Think of the ugliest woman you can. Brienne is uglier. (Gwendoline Christie was just way too cute for her role.)
How Jaime ultimately achieves his redemption is still open to debate. Most fans appear to favor Jaime dying to save either Jon or Bran. Personally I find that, given that he earned his sobriquet "King Slayer" by killing Aerys, he is most likely going to be redeemed by protecting Daenerys. This is the version I use in my hypothetical synopsis.
As far as I can tell, Tyrion has no counterpart in the War of the Roses. As Preston Jacobs has explained on his channel, Martin has routinely centered his stories around a love triangle. The triangle in this case was to be between Arya, Jon, and Tyrion.
With this knowledge we can then surmise what Tyrion's story-arc was meant to be. The idea of Tyrion as a 'secret' Targaryen is intended to put him directly at odds with Jon, another 'secret' Targaryen. Along with this Arya was going to have a strong interest, if not hero worship, for the Targaryen family. The books no longer show this, but we do get a hint in the show.
How this would have played out is more speculative. Likely one of the men would have discovered his heritage first. Regardless of who, Arya would always favor Jon.
At some point Daenerys would attempt to arrange a marriage between Arya and Tyrion, probably because he asked. Arya would reject this utterly and Jon would support her.
This would lead directly to a fight between Jon and Tyrion. It should be clear that both Jon and Tyrion would have become dragonriders before this. There would have been a dragon battle in which Tyrion would come out the loser.
He then would betray Dany and abandon her to go join his brother, Jaime.
Martin's reasons for changing this are complex. First, the dynamic of Jon being the true true born son and Tyrion the bastard was probably a bit much. Second, and more importantly, Martin became far more interested in Tyrion's relationship with Tywin. It is easy to guess that had the original conception held sway Tywin would likely not have become such a central character. The more Tywin the less love triangle.
What needs to also be remembered is that when Martin wrote ASOS it was meant to be near the end of the story. AFFC was going to be the last book in the series, until it wasn't. Tyrion doesn't even appear in AFFC at all. After killing Tywin Martin no longer had a purpose for Tyrion to fulfill. It has been noticed how Tyrion's presence in the story falls dramatically after ASOS. Martin has been trying to keep Tyrion relevant, not unlike the show. In his conversation with Faegon Tyrion manages to persuade him to go west to Westeros rather than east to Dany.
I'm guessing that this conversation was originally going to be between Dany and Faegon instead. This helps keep Tyrion relevant and removes an unnecessary plot line from the story.
Tyrion was and will eventually become Dany's Hand. But if he ultimately betrays her is still up in the air.