By 1919, Marshall wouldn't be much more than a caretaker, and he'd have ll of the WIlson administration baggage to haul behind him: therefore, I don't see him getting any real shot at the 1920 nomination.
Backing up: the widowed Wilson suffers that same stroke during the barnstorming tour to pump up the League as he did in OTL. He's hustled aboard the trains, which makes haste to Washington, where rumors are already swirling. SecState Lansing, Senator Gilbert Hitchcok of NE, McAdoo, Baker and others huddle to gain a consensus on Wilson's status and what should be done next. For the time being, the public is fed the story that Wilson is suffering from "exhaustion" from his speaking tour, and needs total rest, as prescribed by his doctor.
The provisions of the Constitution at the time make it evident that in the case of presidential inability insofar as duties go, those same duties and powers devolve upon the vice president, but the same passage is silent as to who is entitled to make that determination. Baker recommends that the cabinet as a whole draft a letter to the vice president (deliberately excluding Marshall from the decision) saying that it is the sense of the cabinet that the president is unable to discharge his duties, and that as such, he, Marshall, is called upon to declare the presidency vacant by inability, and to assume the office. Lansing vetoes the suggestion, pointing out that in his opinion, that smacks of a coup d'etat, and could provide an unpleasant future precedent if a rift developed in an administration. Finally, a compromise is worked out that doesn't explicitly violate the Constitution, doesn't really please anyone, and doesn't fail to work: the nation is told that Wilson has suffered a stroke that has left him temporarily incapacitated, and that according to the Constitution, Vice President Marshall will act as president. He will not take the oath of office, but will sign any and all documents as "Vice President, acting for the President" with countersignatures by the Secretary of State and the Attorney General.
American government drifts through the balance of 1919 and 1920. Without any real champions on Capitol Hill, the League of Nations treaty is defeated. At the 1920 convention, vice president Marshall gets barely a nod, while in a supreme case of irony, William McAdoo becomes the Democrats' nominee, tapping James Cox of Ohio as his running mate. Meanwhile, the Republicans nominate Frank Lowden of Illinois for president, with William Sproul of Pennsylvania as his running mate. The Republicans campaign on a premise of replacing the "broken, drifting Wilson administration" with a firmer, stronger hand to guide America "through these post-war shoals, free from the snares of foreign entanglements". Lowden defeats McAdoo handily.
Wilson is quietly and gently moved from the White House to a suite at the Soldiers' Home in the District. He passes away in early 1922.