Edward's abdication was because of public opinion and largely due to the actions of the Stanley Baldwin, who convinced the king that moral standards could not allow to marry her. Marrying a divorcee is not a sin of any calibre in the CoE.
Not for a common member, true. But for the head of the church it was a grave issue.
Equally I think you are making a category mistake by assuming the religiousity of people in said world.
No, sorry, the people of New England in the early 1800s were not the secular, casually church-going population of later centuries. I would urge you to do some research on the Second Great Awakening as a beginning. Among the churches most deeply involved in it in New England was the Congregational Church, the direct spiritual descendant of the Puritans.
Finally I don't see what the big issue about Religion is in this theorhetical New England Union argument, Britain had dissenters and that's all New England puritans would be, they could even have special recognition like the Prebyterians of Scotland. I'm sure the thousands of Episopalians in New England wouldn't mind being reunited with the CoE either.
What few thousand there were in New England. During the Revolution, "[o]ut of 55 Anglican clergy in New York and New England, only three were Patriots, two of those being from Massachusetts," according to the church-written history in wikipedia. (It's copiously and accurately footnoted, so I have little hesitation in quoting it here.) One problem Anglican clergy faced during the Revolution was that they had to take an oath of loyalty to the Crown to be ordained, and the Book of Common Prayer included mandatory prayers for the welfare of the King. In 1776 the Continental Congress made oaths of allegiance and prayers for the King acts of treason. And in New England, most Anglicans were Loyalists; many of them were among the 80,000 Loyalists who left the country during and after the Revolution. "By 1790, in a nation of four million, Anglicans were reduced to about ten thousand." Most of them were concentrated in the South.
So there would be no strong push from the Episcopalian side of the aisle in New England for reunion with Britain in the first half of the 1800s.
I will repeat, religion would be only one of many factors standing in the way of reunification, but it would be a major factor, far more so than it would be today, or even in the latter half of the 19th Century.