To start off, most British colonists considered themselves to be loyal British subjects almost right up until the outbreak of war in 1775 (and for some, even later than that). In fact, there's a prominent idea first coined by historian John Murrin called the "Anglicization" thesis: basically, this thesis argues that colonists became more British (in terms of culture, commerce, identity), not less, over the course of the eighteenth century. So there were fewer uprisings in colonial America than you might think. Most colonists were extremely committed to the idea of being British, until the very eve of the Revolution.

Still, there were several American "rebellions" against imperial authority, though independence wasn't usually their leaders' ultimate goal. Bacon's Rebellion is one of the earliest. In this conflict in Virginia, Nathaniel Bacon led an attempt to overthrow Governor William Berkeley, mostly because Berkeley refused to allow Bacon and his followers to seize the land of the neighboring Susquehannocks. However, Bacon argued in his "Declaration of the People" that he was acting on behalf of the English king, Charles II. So this uprising was not presented as a rebellion against the British crown, but rather, as an attempt to curb the power of one allegedly "tyrannical" governor. Bacon did succeed in taking Jamestown, but died soon thereafter, and his movement fizzled out soon after his demise.

The other major American rebellions that come to mind are the ones surrounding the Glorious Revolution. When William of Orange invaded and took the British crown with Parliament's permission in 1688-89, America's colonial governors were left in a precarious position. They had been appointed under the ousted king, James II, and it was not clear whether their positions were voided upon the king's exile. James II had also taken an authoritarian approach to governing the colonies: for instance, he revoked the individual charters of the New England colonies (Massachusetts Bay, Connecticut, etc.), and instead created a single, centralized colony called the Dominion of New England. Colonists in this new dominion had very few of the rights that they had enjoyed previously--they could no longer hold town meetings, or retain their landed property, or have a say in their taxation. So when news began trickling back to America of the Glorious Revolution, colonists in both Boston and New York took advantage of the situation to stage rebellions against their unpopular and suddenly vulnerable governor (Sir Edmund Andros) and lieutenant governor (Francis Nicholson). As with Bacon's Rebellion, however, the leaders of these uprisings explicitly claimed that they were operating on behalf of the British crown. In ousting these Stuart-aligned governors, they claimed that they were acting just as their brethren in England had done, and they pledged their allegiance to Britain's new monarchs, William and Mary. The British state did not always take these proclamations at face value, however: in fact, the leader of the New York rebellion Jacob Leisler was eventually sentenced to a grisly execution for his alleged crimes.

Finally, there was one other set of non-American "rebellions" that would likely have been at the forefront of George III's mind in the 1770s: Scotland's Jacobite Risings of 1715 and 1745. These uprisings were attempts to reestablish the Stuarts--James II, and his descendants, who had been exiled to Europe following the Glorious Revolution--as Britain's monarchs . Jacobitism reached a fever pitch in Scotland following the Act of Union in 1707, and there was always an element of Scottish nationalism at play in these uprisings: many Scots used Jacobitism as a way to advocate for an independent Scotland. After some notable military victories, including the capture of Edinburgh, the Jacobite Rising of 1745 was successfully halted because of a brutal British military campaign led by George II's son, the Duke of Cumberland. A few decades later in the 1760s-70s, George III and his ministers like Lord North used these uprisings as a blueprint for how to deal with the American colonies. In their thinking, Scottish rebellion had only been curbed with a militant, heavy hand, and North America's rebellious spirit would best be halted by similarly militant methods (ex. the "Intolerable Acts" and the military occupation of Boston.) Obviously, this proved to be a serious error of judgment.

In sum, there were a number of rebellions in colonial America, but none of them really qualify as "independence movements." The closest thing to an independence movement during this era was Jacobitism, which was rooted not in America, but in Scotland.

Sources:

Anglicizing America: Empire, Revolution, Republic, edited by Ignacio Gallup-Diaz, Andrew Shankman, and David J. Silverman (2015)

James Rice, Tales from a Revolution: Bacon's Rebellion and the Transformation of Early America (2012)

David Lovejoy, The Glorious Revolution in America (1987)

Allan Macinnes, "Jacobitism in Scotland: Episodic Cause or National Movement?" The Scottish Historical Review (2007): 225-252.