It seems to me that the most interesting time is for the events of the summer of 1915 to spiral out of control. This will effect the election of 1916 and so may cause the maximum amount of difference to the outcome of the war.
Essentially, we need a couple more ships sunk and preferably one of them to be American flagged.
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POD: A German reserve corps is detained from its ordered position on 21 April 1915, near Ypres. The next day these troops are decisive in capitalizing on the advantage gained in the gas attack. The large-scale slaughter of Canadian troops defending against the advance made headlines in English speaking news papers. [OTL, the Germans lack the reserves; here a clerical error delivers them momentum on the Western front.]
This will have the effect of undoing Allied plans for offensives during the summer, as they must respond to the German advance (which grinds on through May and June). Thus the backdrop to the sinking of RMS Lusitania on 5 May 1915 is one of German advantage. British criticism of President Wilson's refusal to take action is more acerbic and combined with the impassioned pleas of Canadian Prime Minister Robert Borden, citing the memory of Ypres (he pronounces it correctly, not "Wipers"). Among the domestic critics of President Wilson is former President Teddy Roosevelt.
Hopeful of a break-through in the West and obsessed by the idea that the Empire might desert Britain (there is much dissension in Canada over the defeat at Ypres, now fiften miles within German-held territory), Kaiser Wilhelm does not beleive it necessary to curb U-boat acitivity during the summer. The sinking of the Arabia in August does give him pause. His declaration rescinding unrestricted submarine warfare came on September 7, 1915--a fateful day in American and later world history.
On that evening, the former American Line steamer SS Ohio, then operating under the International Mercantile Marine Co., a consolidation of several lines (and a controversial trust) is mistakenly targeted by a German U-boat. The news of the change in German policy and of the sinking of the Ohio hits the American press on the same day. What had been an outcry becomes a fury: even some members of German communities spoke out, particularly in Pennsylvannia and Ohio (later scholars have debated their motives, pointing to fears of persecution rather than patriotism). Wilson insists that diplomacy must prevail and again begins protesting to the German government. The Kaiser is uncertain how to balance American neutrality. British and French forces have finally overcome the unsettlement of the summer and have launched a double-offensive, but the Kaiser remains certain that a breakout is at hand. When the British attack on [insert name of mis-pronounced town] fails, he announces that Germany will pay reparations to compensate IMM.
After tumultuous press treatment, Wilson (begining to prepare for the 1916 campaign) begins to accept the offer "in the interests of peace, which above all else, must guide our affairs." Republicans in Congress, particularly Progressives, decry this corrupt bargain between barbarian Huns and a corrupt trust. They press for punitive damages--at least--if not outright war. TR begins speaking about the need to stand up to "pirate diplomacy." Wilson begins negotiations with the German minister in Washingtion.
Things continue over the winter and spring of 1916. In Europe, the German drive in the West has languished since September and the eventual partial success (in holding and partial gains) of the Allied offensive. In America, in dark rooms and the alleyways of power, talk is brewing between Teddy Rooselvelt, his Progressive Party, and the Republicans. Things come to ahead in the Summer of 1916: rather than force action against the French at Verdun, the Germans focus on the British in Picardy, determined to breakthrough to threaten the Channel and Paris. After relentless artillery bombardment, they succeed! They eventually take Abbeville, leaving a pocket of British forces trapped on the Channel coast.
News of this defeat reaches America on the eve of the Republican convention. Without a standard bearer for four years and with the bitter election of 1912, the Republican convention has little consensus. Some favor Charles Evan Hughes, the sitting Supreme Court Justice, for his impartiality (meaning he has been able to say nothing to offend anyone). Yet his very neutrality is ill-suitied to the times: America's neutrality in the war looks more and more like cowardice, some say. And then comes news from the Western Front--the stalemate is over, breakthrough at last, German power seems inexorable. When the news came into the convention hall in Chicago, a wave of shock strikes the delegates. The hall is silent. And then a voice from the balcony: "Mr Chairman, may I have the floor..." In American legend, it is told as a whisper--the mythic soft-spoken words. In reality, it is more like a thunderclap. Teddy Roosevelt's oratory never lacked in volume. And now it's effect is manifest: America must breakforth from its contiental fortress and take on the mantle of world leadership. The fate of Europe, of Christendom, the fate of Freedom itself hangs in the balance.
The Progressive Party, TR proclaimed, would support any candidate nominated by the GOP on a platform of confrontaion with Imperial Germany. The convention's response TR professed to be a surprise; critics called it demagoguery and war fever. Some proclaimed war fever to have still all partisan battles (they forget the old guard insisted on a vice-presidential candidate they could control in the form of Thomas Burton, Senator from Ohio). In any case, the answer seemed clear: Roosevelt won the nomination (which he profusely maintained he had not intended to seek) on the ballot taken immediately after he yeilded the floor.
The campaign that followed was one of the most hard-fought in American memory. Roosevelt campaign promised "confrontation" as near to war campaign as any. Wilson stoutly defended his legacy: "He kept the country out of war." Meanwhile, the American people read avidly of the war in Europe. In France, Britain changed tack: rather than continue the policy of counter attack in pursuit of an elusive breakthrough, Aleaxander Haig chose instead to "bleed the Germans white." Trenches were reinforced with concrete; bunkers constructed--ironically, transfroming Allied trenches (until then disordered affairs) into much better counterparts to the Germans own in an effort to keep the men from retreating. Success came from an unlikely source: the stalled campaign in the Dardenelles. Initially postponed--by frantic cable--the invasion of the Ottoman Empire proved a rousing success. After the German success on 22 April (what was then thought a breakthrough, though nothing compared to the Abbeville offensive), the British called off plans for the hastily prepared invasion of the peninsula on 25 April. Long the source of allo-hisotorical speculation, the additional month allowed Imperial commanders to better scout landing sights and to better accomdate the forces from the Dominions (after the catatrophe of the Canadian Brigade). Each success of the invasion was instantly telegraphed to the British Embassy in Washington for propaganda purposes. [1]
Wilson continued to decry Roosevelt's war fever, citing the casualty reports from the Western Front. Roosevelt averred that "confrontation" was required: if the Germans wanted war, he would give it to them. Thus while most people associated Roosevelt with the likelihood of war, Roosevelt himself made clear that Speaking Softly would still remain an important part of Big Stick diplomacy. Winston Churchill's letter remarked on the significance of the moment: never before in history had a nation contemplated a question of war or peace in such an open, democratic manner. "Fitting," he wrote, "that the safety of democracy should be assured by the practice of democracy." By November it was clear: the electoral math was clear, with the Republicans enjoying the refound support of the Progressives, (especially in Ohio and parts of the West), Roosevelt became the second President in US history to return to the White House in non-consecutive terms.
Nevertheless, there were four months until Roosevelt would take office. Affairs in Berlin were tense: with Roosevelt in office, however, there was little point in restraint. And yet, the United States had still to declare war. Even then, they had little of an army and might provide little weight to the actual fighting for some time. With the threat of renewed support to Russia through the Black Sea and the growing burden of the blockade and the war economy, the necessity was clear: the gloves came off. An all-out offensive against Paris and unrestricted submarine warfare.
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[1] Historians note that these months saw the begining of a correspondence between First Lord of the Admiralty Winston Churchill and Roosevelt. It was this friendship that was memorialized by the friendship between Teddy and Winnie, the Pooh bears, in AA Milne's children books, about the battle between the creatures of the Hundred Acre Wood and the Heffelumps and Woozles (i.e. the Germans and the Turks).