Note: Characters are the ages they were in the show.
Also, I should mention near the start that Woodrow Wilson was in many ways a deeply flawed man and I'm not endorsing everything he does, says or thinks.
Woodrow Wilson found himself in a strange new world indeed. He'd gathered that the woman that had knocked on his door that morning was his mother, and what was more, she was the Queen! The Queen of a kingdom that for all his study of history, and he'd used to teach it, he had never heard of. And surely by some means beyond a human's mind to grasp that he could only conclude must be a part of God's mysterious plan for him, he wasn't in 1919 anymore. No, this reminded him of the old Feudal Europe. Of England in the age of Knights and Chivalry. But he was fairly certain that this was not England. He'd never read of a town called Winterfell when in his days as a student at Princeton, he'd been so absorbed in Macaulay's history that he'd skipped classes to immerse himself in it's multiple volumes.
Certainly England's capital had never been called "King's Landing." So where was he? He'd concluded on the evidence he had so far from brief conversations with his new "family" that he was no longer even on Earth. At least not as he knew it. This was not Europe. It's inhabitants called this continent Westeros. He was lost in the the past perhaps, but the past of some other world, not the world he'd lived in and fought so hard for.
But what was certain was that he was the Crown Prince of whatever kingdom this was. Or kingdoms. Apparently there were seven of them. Perhaps he was heir to some sort of union of kingdoms akin to the Austro-Hungarian Empire that was in the process of being dismantled into independent nation-states back in his own world.
He knew he'd needed to find some books about the history of these lands so that he could get his bearings. He'd managed to obtain one from his "uncle" Tyrion who seemed to travel with a small library of his own. The dwarf (Yes, really!) had been wary when "Joffrey" had asked to borrow some books on history, but had agreed to lend him someone called Archmaester Glyldayn's History of the Targaryen Kings of Westeros.
Before he could get very far in his reading though, he was expected to attend a banquet in honor of his "father," the King. Robert I of the House Baratheon was paying a visit to what seemed to be the northernmost of his kingdoms. There had never been a Baratheon dynasty in old Europe, he was sure of that.
In the banquet hall, Woodrow and his "siblings" were seated in a row opposite the Stark children, excluding one boy who was apparently too young to attend, and including Sansa Stark, the eldest daughter of their hosts, Eddard and Catelyn. He was startled when she was pointed out to him as his betrothed. She was a pretty girl. Auburn haired, and ivory skinned and sweet. But she was only 13 years old. Woodrow was- well he had been 62 years old in September of 1919. Technically he was now a teenaged boy. Yes, in a way he was a young man again, but he still didn't feel comfortable being betrothed to such a young girl. Furthermore, his heart ached when he thought of Edith back home. Poor Edith. What had become of her if as he suspected, he'd died in his own world? Her heart must be broken. It pained him to think of poor Edith.
In fact, without Woodrow to lead his party in the League fight he suspected that the Republicans would end up breaking the heart of the entire world. What those reactionaries didn't understand was that America had a great God-given destiny. She was meant to be a leader in the efforts to secure a just and permanent peace for all the peoples of the world. But how could America be what she was meant to be if Lodge and the others, irreconcilables and "mild" reservationists alike, considered that they had no moral responsibility towards a world they seemed to want only to retreat from? They didn't realize that if they refused to act, the whole victory over Germany would be undone and everything so many brave American boys gave their lives for would be lost.
But that was his old world. There was nothing he could do for that world any longer. He was a stranger in a strange land. Why had God seen fit to take him out of the great struggle and place him here? What was he to do?
He recalled the opening lines of the poem he'd carried with him for so long.
If you can keep your head when all about you
Are losing theirs and blaming it on you,
If you can trust yourself when all men doubt you...
Well. Well, he'd keep his head, that's what he'd do.
And that most of those who knew him seemed to doubt this Prince "Joffrey's" capability to one day rule his father's kingdoms was perhaps an understatement. At least judging from the way his siblings were utterly terrified of him. The way his mother spoiled him, and the way his uncle seemed shocked that the boy would ever voluntarily pick up a book. But Woodrow was here for some purpose and whatever it was, he would fulfill it. If he was to be a King he would do his utmost to lift his subjects to the highest levels of living possible and then one day he would teach Westeros what it was to elect good men.
In any event, the Stark children seemed to be enjoying themselves. The eldest, Robb, who had the same auburn hair as his sister, was happily talking with a darker haired boy sitting near him while the youngest two Starks were laughing amongst themselves at some private joke. Sansa herself was giving him furtive glances as he ate his meal. At one point he caught her eye and she blushed and looked down, but he gave her a reassuring smile and she smiled back at him.
"Now, there's no need to be shy around me," he chided her gently.
Her face reddened again when he spoke to her.
"No. Of course not. I'm sorry, my Prince."
He laughed. "There's nothing to apologize for. And you may call me Joffrey."
He was just beginning to get to know his bride to be (no matter how awkward he felt about that) when a glob of food hit her in the side of the face and slid down to ruin her dress. Her younger sister, Arya, had flung it at her and was laughing uproariously. For a moment a wistful look came over Woodrow as he was taken back to the days when his own daughters were young.
"I hate you! You always ruin everything!" Sansa shouted at her sister and ran from the hall.
At that, Catelyn Stark signaled to her eldest son, who seemed almost as amused at the situation as Arya herself.
"Alright, time time for bed," Robb said with a smile as he picked Arya up and led her out of the hall.
Woodrow resolved that he would talk to Sansa and make her feel better the next day but for now he directed his attention over to the high table. The King and Queen were holding court with various Northern nobles. King Robert seemed to be enjoying himself at least, laughing merrily with his old friend as he gorged himself on all manner of roasted meats and downed great draughts of ale. His "mother," Queen Cersei, was of a different cast of character altogether. She seemed mostly aloof from the people around her, only occasionally favoring one of the northerners with a few brief words or a smile that didn't reach her eyes. Amid the feasting, a singer strung a harp and sang sweet songs that definitely reminded Woodrow of Medieval or Renaissance music he'd heard performed.
Eventually Woodrow decided to step outside for a moment to get some fresh air. As a young man again the cold air only invigorated him as he walked out into the courtyard. Suddenly he caught the sound of voices in the dark.
"Remember this, boy. All dwarfs may be bastards, yet not all bastards need be dwarfs."
With that his uncle turned and began to saunter back to the banquet, whistling a cheerful tune along the way. He stopped when he saw "Joffrey" standing there alone in the night.
"Ah, nephew. Bored of the feast already?"
"Who was that you were talking to uncle?"
"Lord Stark's bastard. Lady Catelyn thought it best he not attend the banquet. Thought it might offend us," he said amusedly. Then his face turned serious. "Have you... read any of Archmaester Glyldayn's history?"
"Oh, yes. I didn't have time to get very far of course. I'm still only on Aegon's Conquest. But then, I've never been a fast reader you know, uncle."
Tyrion seemed to study him. "I've never known you to read at all."
"No... Well no, not really, I suppose not. But I'm to be king one day and I thought it was time I had a thorough understanding of the past and how my predecessors governed so that I can give these Kingdoms the best possible future. Nobody can be a true man of the world who knows it only as it is in the present day. Having said that though, I would like to understand our current political situation better..."
Tyrion raised an eyebrow. "Is that so? Well I don't have the faintest idea what's gotten into you nephew, but if you do manage to get through the book, I can recommend The History of the Greater and the Lesser Houses. You can probably find it in the library here at Winterfell. I'm sure every noble house on the continent has a copy. It surveys their histories from their beginnings until shortly after your father's reign began. It should give you the basics on the different types of people you're to rule one day, although the prose leaves something to be desired."
"Ah! Thank you, just what I needed! I'll check the library tomorrow. For now, I think I'll go to my rooms and read a bit more on Aegon before bed. Good night, uncle."
Tyrion stared dumbfounded as Joffrey walked away.