Having looked at the Monmouth Rebellion before, I'd have to say that the most plausible (IMO) way of getting it to succeed is to butterfly some of James stillborn daughters away and have Mary of Modena "give birth to a son" shortly before Monmouth's invasion actually begins.
Monmouth lands and is immediately heralded as the saviour of Protestantism as large numbers flock to his cause. Though James II has little to fear from this organized rabble, his concern grows as his predominantly Protestant army begins to defect in increasingly large numbers as Monmouths rag-tag army begins it's march on London.
Though James II does attempt to defend London with loyal Irish troops, rumors of "An Irish invasion" prompt the general populace of the city to rise up en masse and force James II and his family to flee. Upon hearing of James II's retreat, Monmouth takes an advance column into the city and strikes a deal with Parliament. In exchange for legitimizing his father's (Charles II's_ affair with his mother and crowning him James III, Monmouth agrees to surrender a great deal of power to the Parliament (Basically the same thing William III had to do.) By doing this, Parliament has not only put a Protestant King on the throne but also quelled a potential revolt and threat to their authority.
Meanwhile, across the English Channel in the Dutch Republic, Wilhelm III is far from pleased. Against all odds, Monmouth had succeeded in effect stealing the throne which Wilhelm had been maneuvering to acquire for himself. Yet all was not lost, for Wilhelm III was indeed a great deal more legitimate than the new King of England and (technically) Scotland. Only the agreement of James III and Parliament to join Wilhelm's Grand Alliance prevented Wilhelm from causing problems for the shaky regime.
James III will undoubtedly move quickly and consolidate his rule in Scotland before moving onto Ireland. There he'll most likely engage his uncle yet again in one of many Jacobite Rebellions. After that, it's anyone's guess...
Probably no Hannoverians though...