PART I
I'll open with this damning quote from John Milton Cooper, widely viewed as the preeminent modern academic biographer of Wilson, which rather bluntly sums up the overall effect:
"...[Wilson's stroke-effected] emotional imbalance and skewed judgment blocked a more constructive outcome [for the League of Nations treaty]. At times in the first three months of 1920, he did seem to verge on mental instability, if not insanity. [His staff and physicians] did the best they could by their lights, but they were frightened, limited people who should not have been trying to keep Wilson's presidency afloat. He should not have remained in office. If he had not, the League fight would have turned out differently, and the nation and the world would have been better off."
I've previously gone through an extensive walkthrough of presidential disability up through 1950 - and yes, I still need to get Part III on Eisenhower, JFK, and the 25th Amendment up one of these years - but I'll start my own longer answer to this with the overall summary on Wilson that I provided there.
Here's a basic outline: Wilson stroked out in October 1919 with the entire left side of his body paralyzed, including a near fatal blockage of his bladder. Over the next six months, as he refused to see almost anyone including any members of his Cabinet, the Vice President, or Congress, his actual condition was entirely unknown except for the tiny handful of people allowed in his presence. His Secretary of State Robert Lansing tried to run the Cabinet in the meantime and get Vice President Thomas Marshall to take action; Marshall refused, partially because there was no clear legal authority for him to do so. While he regained some function, Wilson never really recovered enough to do much afterwards, leaving what was effectively a power vacuum at the top during the last 18 months of his term, made far worse as it was one of the most consequential recent periods of both European and American history.
But worst of all from a continuity of government perspective? When Wilson recovered enough to at least make minor decisions, one of the first he made in February was to fire Lansing for the audacity of running the Cabinet without him. In the meantime, as it had during Garfield's extended decline, Congress began debating what actions it could take. During Garfield, there had been a debate about what 'inability' - the sole reference in the Constitution on Presidential disability at that time - actually meant, and there was no conclusion on either the definition or who had the right to make that decision - but some agreement was reached that perhaps in whatever those circumstances were the Vice President should act temporarily.
During Wilson, it eventually got as far as dueling proposals to authorize either the Supreme Court or the Cabinet to determine if the President was disabled, with the sticking point this time being that if the determination was indeed made, would the Vice President would even be constitutionally allowed to do so on a temporary basis or would they be required to serve out the rest of the Presidential term? Or did Congress even have the power to legislate most of what was being proposed to begin with, let alone any chance of a consensus on what needed to be done?
This set of Constitutional brambles was one reason why Marshall hesitated despite likely being one of the tiny handful who probably was briefed quite early on Wilson's true condition, but to explain this properly, we need an extensive look at Wilson's health history prior to his October 2, 1919 stroke.
PART II
So let's start our walk through this with a brief dive into cerebrovascular accidents (CVAs) - aka strokes - and how physicians and the public have viewed them both in the present and the past.
From the perspective of the 21st century, we now have a pretty good handle on the risk factors for ischemic strokes - the ~8.5 out of 10 strokes that are caused by a blockage of blood flow to the brain - as well as those for hemorrhagic strokes, caused by bleeding in or around the brain. I won't go into detail on the latter since it's not what Wilson suffered from; even at the time, a neurologist explained to Edith Wilson that if her husband had one, he'd have have almost certainly died on the spot.
What's generally regarded as the second most important risk factor for ischemic strokes is hypertension, which increases your baseline chance of having one by about threefold. Hypertension causes a plethora of other health problems besides stroke risk, but for the first half of the 20th century it was generally believed as best medical practice to not treat (with the limited options available) hypertension until it got to levels around what we'd consider today as hypertensive emergencies, or over 180/120. As I've written before, up until the 1950s and the Framingham Heart Study, it was generally believed by most physicians that higher blood pressure was simply a natural part of aging rather than being a contributory factor to all sorts of diseases.
But the single most significant risk factor for a stroke? Having had one previously. This has led to a significant number of physicians delving into Wilson's medical history, and it is now very clear that the October 2 stroke was far from his first; in fact, his very first stroke - possibly a TIA (transient ischemic attack, basically a ministroke) - may have been at the tender age of 35 in 1891, when he wrote to a friend about having some sort of temporary illness related to his nerves that caused him to have difficulty holding his pen.
Why is something related to "nerves" telling? It's because the contemporary view of the etiology of strokes was very different than today. If you want a deeper dive into the history of apoplexy - the more modern common vernacular of "stroke" appears to have gained traction shortly after Wilson's death - there's a thread here including a contribution by /u/BedsideRounds. A simple summary is that medicine at the time had established the physical damage that a stroke causes in the brain, but had only tangentially connected hypertension among other causations, and that only among a few specialists. Instead, the widespread perception both among physicians and lay persons was that a stroke was something mostly caused by mental exhaustion, fatigue, stress, and overwork.
Now, Wilson was a bit of a hypochondriac - albeit not in Madison's league, who complained of being near death all sorts of times until as mentioned in the link above when he actually was close during his Presidency as mentioned above - but most of his other early complaints seem to revolve around gastroenterology problems; for a couple decades he sometimes used a stomach pump connected to a rubber hose to 'rebalance his acids' with saline and coal. While this was a common treatment of the era, it doesn't seem to have done him much good, and continued until he moved into the White House and finally got convinced to stop. In fact, this came out of the first consultation he had with then-Lieutenant Dr. Cary Grayson, medical officer of the Presidential Yacht Mayflower, who impressed Wilson enough in debating with him what to do about his health that he shortly thereafter invited Secretary of the Navy Josephus Daniels over for lunch to instruct him to assign Grayson as his full time White House physician.
While we don't know for sure about the 1891 incident, a much more serious one took place 5 years later in 1896 that is better documented and in the context of what we know now, far more disturbing especially considering he was still only 40 at the time. Wilson by that point was now the very prominent chair of what was essentially the Political Science department at Princeton and is busy unintentionally proving that publish or perish can consist of both; his wife Ellen mentions to a friend that he "almost killed himself [making $4000 extra from books above his $1500 salary]" that year.
What she was referring to was her husband losing feeling in his right arm and hand and suffering considerable pain from it. He consults with some of the most prominent neurologists in the country (conveniently located in Philadelphia) and is told that he has "writers cramp" or "neuritis" and some rest is recommended; modern MDs have suggested that it was an occlusion of his left middle cerebral artery. Fortuitously, he's already charmed a wealthy neighbor who previously offered the couple a trip to Great Britain on her; this turns into a two month rest cure without Ellen. He regains full use of his right arm during the voyage over, learns to write with his left alongside that (Wilson has long been ambidextrous), and largely thinks nothing of the whole incident until another 8 years pass when his right hand is temporarily weakened once again in 1904. His brother in law writes that he felt Wilson's personality changed after this; he became more driven and focused on national politics rather than political theory.
(Part II continued below)