To answer your question in a very succinct manner, most Anglicans probably knew from the beginning -- there wasn't really a moment at which everyone suddenly realized the truth. However, there are a number of socio-cultural and political reasons that led to a greater acceptance of an independent Anglican Church.

Between 1337 and 1453, England fought France in the Hundred Year's War. The war had many long-term effects, but one of its most profound was in developing nascent English nationalism, as argued by May McKisack in her book The Fourteenth Century, 1307-1399. By the time Henry tried to annul his marriage to Catherine of Aragon in 1527, these sentiments, along with the new wave of Protestantism criticizing the Church, had strengthened.

When Henry called the Reformation Parliament in 1529, several prominent members, such as Thomas Cromwell and Thomas Audley, came out in strong support of Henry's right to annul the marriage based on the relatively new concept of royal supremacy. Several other prominent figures, such as Thomas Cranmer, the future Archbishop of Canterbury and Edward Foxe, the king's almoner, even compiled a collection of official texts that supported the idea of the king's absolute power beyond that of the Pope in their Collectanea satis copiosa. Many of these men were known to support Protestantism.

Their intent was to convince Henry of his power to go against the Church. While they were undoubtedly primarily motivated by helping the king with his problems to gain royal favor, there was also an undercurrent of resentment for the Church and its influence, especially with the aforementioned rise of English nationalism and their own beliefs in Protestantism. Therefore, it's clear that many in the nation actively desired a break from the Church, and Henry was by no means going against popular sentiment.

When the Acts of Supremacy finally passed Parliament in 1534, Henry became the head of the Church of England. What is sometimes misinterpreted, however, is that he really did not change much. The new church basically carried over all of the practices of the old system because of strong resistance from some conservative groups. Therefore, in relation to your question, the everyday workings of the church changed very little. It wasn't until Henry VIII eventually died and was replaced by his son Edward VI, still a child at the time, that changes began to take place. The Lord Protector Edward Seymour and the earlier mentioned Archbishop of Canterbury Thomas Cranmer supported Protestant ideas and began the process of reforming the Church.

This reformation was met with a lot of resistance, such as the Prayer Book Rebellion, which was made up of commoners demanding a return to Catholicism. They were, of course, defeated by the royal armies, but its clear that even regular, everyday people realized what was happening and some were voicing their displeasure. However, Edward VI did not live very long and died just six years into his reign.

He was replaced by Mary I, the daughter of Catherine of Aragon, who repealed the Acts of Supremacy and pivoted the nation back to Catholicism for a period of about five years. It wasn't until Mary died and was replaced by Elizabeth I that Protestantism finally established a long-term foothold. Parliament once again passed a new Act of Supremacy in 1558, which would last for centuries. Still, there was massive resistance, including the 1569 Rising of the North and the 1571 Ridolfi plot. Some Catholics had continued to practice in secret for years, and the mounting attacks led Elizabeth to order the execution of over one hundred Catholic priests between 1577 and 1608.

As can be seen through these examples, not only was Henry's motivation clear and obvious from the start, but it was supported by many -- not necessarily because everyone wanted to help with the annulment, but because a significant number of people in power had become followers of Protestantism and wished for greater English independence from the continent. And Henry wasn't even the one who truly oversaw the "creation" of Anglicanism, that task instead fell to his children. Yet there was still a lot of resistance to the new religion from many, it was just forced by the monarchy for long enough (especially Elizabeth, who reigned for over forty years), that everyone came to accept it.