This is a very interesting question, and one that I hope I am not the most qualified to answer. Most of my experience relates to the material culture of the Southern Colonies from 1760 to 1780 but I will take a stab at your question. If anyone has more information, please add to it.

The American Revolution seriously divided the Anglican Church, but it was not a matter of divinity but rather political allegiance and oath. This idea of the divine authority of the Crown was already out of fashion by the advent of the Glorious Revolution, and the reign of James the II made it clear that Anglican Ministers could not reconcile their allegiance to their King and their conscience.

Almost half of the Anglican Priesthood remained Loyal to the Crown out of their oath of loyalty to the King before God, but it underlies the serious divisions in American society brought over the Revolution. Latest research indicates a rough 3-way split, 1/3rd of the population were Patriots, 1/3rd were Loyalists and 1/3rd really just wanted to be left alone. [EDITED] Research indicates that about 15%-20% were Loyalists.

What happened was a switch, they viewed the breakaway from the Crown as a worthy and holy goal against tyranny. Their broken oath to the King a victim of his mismanagement and led to the creation of a separate American "Anglican" Church. William White pushed for a revised book of Common Prayer and a new Church more compatible with the democratic ideals of the new nation. After several conventions, it became the Protestant Episcopal Church of the United States in 1789 in Philadelphia. We see examples of this even earlier. On May 25, 1776, the Maryland Convetion voted "that every Prayer and Petition for the King's Majesty, in the book of Common Prayer . . . be henceforth omitted in all Churches and Chapels in this Province." Similarly, in Philadelphia, (any reference to the welfare of the King, Royal Family and George III was replaced by references to Congress.

This is not all, Anglican preachers on the other side of the political spectrum shot back, Loyalist preacher Jonathan Odell (a rector from New Jersey) wrote a lengthy, rhyming attack on those declaring indepedence and denounces any preacher that abandoned his oath and charges them with apostasy. In a rather forced rhyme he says "Whilst to myself I've humm'd in dismal tune, I'd rather be a dog than Witherspoon." Referencing the John Witherspoon, the Presbyterian minister who was a signer of the Declaration of Independence.

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Sources:

-The Role of the Anglicans in the American Revolution, William Warren SweetHuntington Library Quarterly. Vol. 11, No. 1 (Nov., 1947), pp. 51-70

-Book of Common Prayer. England: John Baskerville, c. 1762. Washington National Cathedral Rare Books Library-Book of Common Prayer. London: Mark Basket, 1766. Courtesy of the Rector, Church Wardens, and Vestrymen of Christ Church, Philadelphia-The Case of the Episcopal Churches in the United States Considered. William White. Philadelphia: David Claypoole, 1782. Rare Book and Special Collections Division, Library of Congress-http://www.pbs.org/opb/historydetectives/feature/the-church-and-the-revolutionary-war/-The American Times: A Satire in Three Parts in which are delineated . . . the Leaders of the American Rebellion. Jonathan Odell, London: 1780. Rare Book and Special Collections Division, Library of Congress -

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The English monarch is the “Supreme Governor” of the Church of England. This was a conscious decision, as Christ is the Supreme (and divine) Head of the Church, making the king/queen simply one of Christ’s officials.

If you look at the Declaration of Independence, it follows the pattern at the time of not challenging the monarch, but alleging that he had been led astray by his officials. George III was not the divine head of the Church of England, but merely an official, and so it was possible to conceive of Anglicanism not governed by the Church of England. The Scottish Episcopal Church already existed under this logic, and in fact was used by the Americans as their official source of holy orders.

Source: ipse dixit, as a former Episcopalian turned RC