Frankly, the Battle of Saratoga is a terrible point-of-divergence for an alternate history.
The whole idea of sending an army south from Canada to invade the Hudson valley was a bad one, based on the delusional belief that a majority of the American colonists were Loyalists who would rise up against the dominant pro-independence faction as soon as a British army appeared. In practice, General John Burgoyne’s plan was crippled (though he didn’t know it) by the fact that his commanding officer, General Sir William Howe, had no intention of sending his army north to meet Burgoyne in Albany. Before leaving for his attack on Philadelphia, Howe specifically instructed his second-in-command, Sir Henry Clinton, not to leave the vicinity of Manhattan. Thus, even if everything had gone right for Burgoyne, the best he could hope for was to reach Albany himself before winter set in, and maintain his army there while his superiors in London decided what to do next. The next year might bring a resumption of the march downriver or an invasion of New England; or it might bring orders to abandon Albany and retreat back to Canada. As it turned out, the defeat Burgoyne suffered was always a probable event, and it became a near-certainty after his advance was halted by Horatio Gates’ army at the Battle of Freeman’s Farm.
So why did Sobel make the Battle of Saratoga his point-of-divergence? Because the battle is so often called the turning point of the American Revolutionary War. (In fact, Richard M. Ketchum titled his 1997 history of the battle Saratoga: Turning Point of America’s Revolutionary War.) It is true that the American victory at Saratoga persuaded a wavering King Louis XVI to commit to a formal alliance with the rebellious colonists. However, absent some total disaster for the Americans, a French alliance was bound to happen eventually; the chance to detach the American colonies from the British Empire was too tempting for the French to pass up. Thus, Saratoga should be thought of as a milestone rather than a turning point.
For alternate history purposes, the earlier Battle of Trenton would make a much better point-of-divergence. Washington’s army was on the point of disintegrating as the soldiers’ terms of enlistment were about to expire. The attack on the Hessian garrison at Trenton was a desperate move that depended on large amounts of luck to succeed. If Washington’s luck had deserted him, his army would have been crushed, and he himself might have been captured or killed, dealing a serious (perhaps even fatal) blow to the American cause.
But the Battle of Saratoga is what I have to work with. The trick is getting Clinton to move north to save Burgoyne, in spite of his orders from Howe. This is how I’ve decided to do it:
* * *
(the story picks up after the battle of Freeman’s Farm on September 19, 1777)
Burgoyne’s campaign, which had begun with such promise, now teetered on the brink of failure. Burgoyne’s men had expected another victory on the order of Ticonderoga. Instead, the rebels had fought them to a standstill. Burgoyne himself had been in the thick of the fighting, and was under no illusions about the cost that would have to be paid if he wished to continue the advance on Albany. At this point, he received word from Clinton, who had written on September 12 that he intended to “make a push at [Fort] Montgomery in about ten days.”
Burgoyne now saw a chance to revive his original plan to catch the rebels between two armies. However, making it happen would require a carefully phrased request. Clinton was neither below him nor above him in the chain of command, so Burgoyne could neither give him orders nor request them from him. Instead, drawing on his skills as a playwright, Burgoyne crafted an appeal for help. “My situation is most perilous,” he wrote. “Unless I receive succor, this great Enterprise must founder, and the work of ending this Rebellion be set back, it may be, years. I may not command it of you, but I do most humbly beseech you to come to my aid with what force you may.” 
Burgoyne had no way of knowing whether his appeal would succeed in bringing Clinton north, but he knew that he had no other hope of victory, so he abandoned his advance and began fortifying his forces along the battle line that had been established on the 19th. Gates, for his part, knew nothing of the exchange of letters between the British generals, and he remained in his own defensive position at Bemis Heights. The two armies continued their defensive postures for the next two and a half weeks. One of Burgoyne’s officers later wrote, “I suppose seldom two armies remained looking at each other so long without coming to action.” 
Clinton was waiting on reinforcements from Britain which did not arrive until the end of September. With an additional 1700 men added to his army, he sailed up the Hudson on October 3. Landing three days later south of Fort Montgomery, Clinton finally received Burgoyne’s plea for help. The dramatist’s words succeeded in persuading the ordinarily cautious Clinton to move north to Albany. Clinton’s army landed south of the town on October 8, then bypassed the city, which had been garrisoned with 2000 rebel militia, and marched north along the west bank of the Hudson. General Israel Putnam attempted to block Clinton’s path on October 12, but Clinton was able to surprise Putnam with a flank attack led by General John Vaughan, and Putnam’s force was crushed.
Burgoyne, hearing no word from Clinton and with his own supplies running low, attempted a flanking maneuver of his own on October 7 that ended with another repulse by Gates’ men. Burgoyne might have managed to disengage from Gates at this point and retreated back to Ticonderoga, but he still pinned his hopes on Clinton’s arrival. He led his army north across the Fishkill River on the night of October 8-9, then dug into a fortified position.
It seemed to Burgoyne that he had lost his gamble when he learned on October 13 that the rebels had surrounded his army. He sent one of his men to Gates with an offer to meet the next day to discuss terms of his surrender, and Gates agreed to a parlay the next morning. During the parlay, however, Gates issued his own terms, which amounted to unconditional surrender. When Burgoyne’s officers learned of the terms, they declared unanimously “that they would rather die than accept such dishonorable conditions.” That evening, Burgoyne learned that he had won his gamble after all when a messenger arrived from Clinton with word of his victory over Putnam. Putnam’s own messengers to Gates lost their way and never made it to his headquarters at Bemis Heights, so he remained unaware of Clinton’s approach.
Since Burgoyne had refused Gates’ surrender terms, and declined to offer his own, the stalemate between the two armies resumed until October 20, when the arrival of civilian refugees from the south alerted Gates at last to Clinton’s approach. Panicking, Gates ordered an all-out assault on Burgoyne’s position the next morning. A series of charges by Gates’ army was thrown back, and casualties mounted among the rebel troops. Gates’ headquarters at Bemis Heights was overflowing with wounded men and in a state of chaos the next day when Clinton’s army broke through his rear lines. Burgoyne, hearing the sounds of battle to the south, roused his own men into a final attack. With hostile armies on both sides of him, Gates fled the battlefield, and organized resistance to the British collapsed. Most of the rebel militia melted away, returning to their homes. The remainder of Gates’ Continental Army troops attempted to flee to Albany, which was still garrisoned by rebel militia. Clinton was able to pin them against the Hudson, and on October 25 Gates accepted surrender terms offered by Burgoyne. Both Gates and his men would be permitted to return to their homes unmolested provided that Gates pledged never again to take up arms against the Crown. Gates accepted, and the last organized rebel army in New York province dispersed.
1. Henry Mitchell. The Battle of Saratoga-Albany (London, 1939), p. 98.
2. “The Journal of Lt. William Digby” from Joanna Brooks. ed. The Face of War: Diaries of the Army of Nations (London, 1956).