Probably some Great War equivalent where the US ends up opposed to Britain. In that era, the Anglo-Japanese alliance is still in effect, so Japan will be involved too, although they probably won't be doing much in the Americas (although tying down a large part of the US's Pacific assets is noticeable). To make sure most of the combat ends up in the Americas, you'd need a decisive defeat in Europe of whoever the US is allies with. Maybe it would be France-Austria-US-Ottomans versus UK-Germany-Italy-Russia (it would take a 19th century POD to get those alliances). After a year or two, the fighting in Europe ends with decisive victory against the US's allies, but the US refuses to quit the war out of national pride (perhaps the British and/or Japanese did something completely unacceptable, like Pearl Harbor/9-11 levels of unacceptable).
In the Americas, perhaps you also have Brazil and Argentina fighting, one on either side. The British and Americans each had large interests in both countries, and Chile as well. But since the US built Argentina's dreadnoughts, maybe the Argies join the US side, and Brazil and Chile are on the British side. Since we want the largest conflict possible, let's have Peru on the American side as well, and maybe Bolivia too, bribed in exchange for a reversal of the War of the Pacific (and some nice loans of money and supplies for the war). Colombia and Ecuador might be brought in on the British side in exchange for resolution to their conflicts against Peru, and perhaps Colombia could also be offered the chance to reconquered Panama. Venezuela will probably be American, since the US will probably do everything they can to deny the British their oil and also offer them that chunk of (British) Guyana they claim. Uruguay might have another civil war between the Blancos and Colorados, one side of which will be pro-Brazilian, the other side pro-Argentine, while Paraguay will likewise probably a civil war regardless of what any exterior power does (it wasn't very stable in the early 20th century).
Central America and Mexico will probably be entirely American, due to the interests of American companies. They probably won't be able to contribute much (although Mexico might be useful in some roles, plus they have oil which won't be sold to the UK or Japan), but Guatemalans and Mexicans will play a role in the invasion of Belize. The Caribbean states--Cuba, Haiti, Dominican Republic--will also be allied to the US, and will also not be able to contribute much except for in some logistics roles for the Caribbean Campaign, which will see the US attempt to eliminate Royal Navy presence or at the very least, secure Trinidad and its oil.
There's still the Dutch and Danish possessions in the New World which are neutral, so we'll have the Dutch enter into the war on the Franco-American side after Germany invades them (as one version of the Schlieffen plan suggested), and as for Denmark, maybe they get bullied into the British side somehow. In that case, the Dutch in Suriname and their Caribbean possessions will probably have a small role to play, but likely will capitulate early on in exchange for being able to keep Indonesia. Denmark adds the Virgin Islands which will probably be an early target for the US, assuming the US is ever able to go on the offensive in the Caribbean. There might an operation in Greenland aimed at sabotaging the cryolite mines, but a full invasion probably won't happen.
I don't see a Russian invasion of Alaska, but there might be some combat in the Bering/North Pacific area between the US and Russia. An Anglo-Canadian invasion of Alaska will probably occur, and probably a conflict where the US will lose assuming it's done early on and the US suffers a naval defeat or two in the area. Conversely, I doubt the US will be able to do much to the Russians--at most they might be able to shell Petropavlovsk assuming they're able to deal with the Russo-Japanese forces (if this is pre-Russo-Japanese War, great, if not, then their coordination will be non-existent) in the area.
The largest theater will probably be in northeast America, in New England and the Midwest. The British will probably deploy a ton of forces there to reinforce the Canadians. The second largest would be the Southern Cone, mostly along the Argentine borders, but also along the Andes (wouldn't be hard to hold the passes) and in the northern deserts. That war will mostly be a South American affair, since I don't see large numbers of European, US American, or Japanese soldiers fighting there--they will all be likely reinforced by the larger powers, so there will be considerable naval action there. The Caribbean theater will be mostly Latin Americans and US American forces (mainly concentrated around the Panama Canal, Venezuela, and Puerto Rico), but probably will have some British forces involved as well, and not just in the defense of their possessions there. I don't think the Japanese would get involved that far from home, outside of perhaps some naval missions.
The US wins in Canada hands down (after a rough start), and almost certainly will be able to hold Puerto Rico and keep her Caribbean allies safe, but the naval war could go either way (and there will be some major losses, especially early on before the US can really gear up for war), especially since it isn't guaranteed the US can keep the Panama Canal. In South America, only Venezuela is likely to come out ahead in this conflict, since the US will have trouble supplying Argentina and Peru with significant amounts of resources compared to the British and Japanese supplying the Brazilians and Chileans. The best chance would be an internal conflict in Brazil which would tie down a large amount of their soldiers and resources, and maybe even lead the Brazilians to bow out of the war in exchange for something minor. South America is going to be an utter mess here. At best, the US boots the UK out of the Caribbean for good, creates some new puppet republics in Canada (maybe even annexing parts or all of British Columbia plus Yukon) and occupies Newfoundland. At worst, the US loses Alaska, the Panama Canal, the Philippines, and probably some Pacific islands, and restores independence to Hawaii and probably Puerto Rico. Canada being occupied is going to be a huge bargaining chip which will probably allow the US to regain anything they lose to the British.
Overall, not the most likely scenario, and a bit contrived in terms of getting every single state who had land in the Americas involved in the conflict, but that would be a great way to have the largest American war possible in the 20th century. By WWII, the US is just too dominant and too influential to really make it a fair fight.