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.
The first answer is pretty simple: they knew immediately.
For some reason, the medias of the last few decades have depicted Henry VIII as this extremely passionate man who would spend his days having sex with beautiful women but who was almost completely disinterested in politics and religion.
In truth, Henry was an extremely smart man who had spent his entire childhood preparing to become a priest (his older brother Arthur should've been the one crowned king, if he hadn't died at fifteen, and it would've been perfectly normal for Henry to enter the church), and enjoyed reading the Scriptures and could quote many passages by memory - it was one of the interests him and Anne Boleyn shared. When he dug out the passages of the Leviticus that say "You shall not uncover the nakedness of your brother's wife; it is your brother's nakedness" and "If a man takes his brother's wife, it is impurity. He has uncovered his brother's nakedness; they shall be childless", he was perfectly aware of the fact that those passages were meant for Jewish people and not Catholics, but relied on both the fact that he had received the title of Fidei Defensor (Defender of the Faith) in 1521 and on the clear fact that, in twenty-four years of marriage, he and Catherine had only had one child who survived to adulthood (and since he was the king, no one had pointed out to him that even the bastard child he had recognized as his, Henry Fitzroy, had always been sickly and died young too).
The English people, however, didn't really care for that: they all liked Catherine, because even if she spoke broken English, she had always done huge works of charity and was known for her strong faith. When she was removed from Henry's court, she had been in England for thirty-three years, and from 1533 to her death in 1536, they kept talking about her as if she was still the queen.
They all knew the reason why there had been a religious shift and Henry had declared himself head of this new, only English church, was because he wanted to marry Anne: the affairs of the court were pretty much known by everybody because gossip used to travel as fast as it does today, even without phones and the internet. They all knew that the king had been interested in Thomas Boleyn's youngest daughter, that he had been courting her since around 1527, that prior to that he had most likely had a relationship with her older sister (and at some point, after Anne's executions, gossip began saying that he had had a relationship with their mother too, but this theory is completely unfounded), and that she didn't want to become his mistress like Bessie Blount had been. The fact that he wanted to divorce his wife was immediately, and rightfully, connected to the fact that Anne had been enjoying his attentions (she would be given important roles within the court, such as organizing trips and events, activities that usually only the queen would take care of. In 1532, Henry also created a nobility title only for her and that allowed her to earn 1000 pounds per year). When she was crowned, sporting a pregnant belly, people would either shout "God save the queen", without shouting the name of said queen (therefore making it clear that they didn't mean Anne), or stand in complete silence.
For the other question: people continued with Anglicanism because otherwise they'd be murdered. In the year following the institution of the Anglican church, Henry made sure to disband churches, execute nobles who didn't want to think of him as the head of this new branch of Christianity, and ordered to burn down buildings and books. The women he married next, especially Jane Seymour and Catherine Parr, actually used their influence to stop him from executing even more people and burning down even more buildings. People didn't have a choice if they wanted to keep on living.
Henry was an incredibly smart but easily susceptible king who wasn't used to being told no, and he was known for turning onto his friends too. He didn't even allow his wife of twenty-four years to hold the hand of their only surviving daughter as she died, he sent one of his closest friends, Cardinal Thomas Wolsey, to die in complete loneliness, he murdered two of his wives and planned on murdering a third one (Catherine Parr, but then he went into a coma and died before giving the order), and got executed even old people part of the same family tree as his. Everybody knew how dangerous that man was, and, well... if he wanted a new church to be able to marry whoever he wanted, so be it.
Sources:
- The Six Wives of Henry VIII, by Alison Weir;
- The Creation of Anne Boleyn, Susan Bordo.