The short answer is concentrated power combined with influence and fear.

To understand the reigns of the children, we need to take a step back and examine the father.

Henry VIII was by all reports a devoted Catholic. Unfortunately he was only the second ruler in a new line in a still significantly divided country. Before his question of succession, Henry was the Defender of the (Catholic) Faith. Catherine of Aragon was a devoted Spanish Catholic. Once the problem of succession came to the fore, Henry became very pragmatic in the creation of the new Church of England. With the assistance of Thomas Cromwell he made, what was essentially, a Catholic church with a protestant faith. By all evidence, it was quite moderate. He needed a divorce. This wasn't a question of faith. Only true believers like Thomas Moore were dedicated to the strictly catholic belief. The English reformation under Henry was hardly dramatic and was easily administered because it required little in the way of actual change. Henry (and Edward after) pillaged the Catholic abbeys of their riches and used the wealth to buy influence and popularize Anglicanism through material gain. Combine this with the circulation of the Book of Common Prayer, it became easy to circulate the new religion among the masses.

Once Henry died, the advisers to the boy king Edward were Protestant radicals, especially the Dukes of Somerset and Northumberland, his regents. Robert Scribner does a wonderful job in his article Popular Magic (I can't find/remember the exact title) where he shows specifically how Anglicanism began to differ radically during this period from Henry's Anglicanism, which could be described as Catholicism-light. The Book of Common Prayer under Edward begins to seriously change the religious landscape of the country and Henry's religion of convenience changed quickly.

The protestants in power knew that Mary could not be allowed to rule. She'd be raised a hardline Spanish Catholic, plus she had the shame of a once great catholic father who abandoned her mother and her faith to atone for. This is why Edward's will was changed from "The male heirs of the Lady Jane Grey" to "The Lady Jane Grey and her male heirs." Radical protestants knew heads would roll (literally) if Mary gained the throne. Which, of course, she did. Mary enforced her Catholicism through military strength and fear through public executions. (EDIT) Mary's cousin was Charles V King of Spain, the Holy Roman Emperor. So she had powerful allies on the continent that were connected by both blood and faith. She didn't fear English separatism like Henry did in the latter years of his reign because she knew she had the military strength to enforce her will on the population. (/EDIT)

Elizabeth was a return to moderate Anglicanism, much like that of her father. This was an easy sell. Most of the people alive remembered Henry's reign and after two short reigns of two radical monarchs, the people were exhausted. Elizabeth used practicality as a tool to unify a divided country.

Now I'm obviously glazing over decades of history, but that's about the best I can do without getting even longer.

IMPORTANT TAKEAWAY: The English Reformation was extremely top-down. Henry used religion as a political tool to secure his power and establish a legacy. His children had to deal with that legacy in their own fashion. It's a very different story than the efforts to reform on the continent.

EDIT: Found the article:

Scribner, Robert W. "The Reformation, Popular Magic, and the 'Disenchantment of the World'". In The Journal of Interdisciplinary History, 23, 3 (Winter 1993). 475-494.

As a follow-up, what would the experience of your average parishioner been over this period? Would they have noticed changes in church practice or sermon content etc?

Part 1.

Adding to /u/BrianFlanagan 's fantastic answer, it is worthwhile to consider the broader picture of Europe at that time.

The reign of Henry VIII coincided with the rise of Charles V of HRE and Spain (and of Italy, of the Low Countries, etc. etc. etc.); and the rise of Francis I of France. It should be noted that both fought for HRE emperorship, a contest that was won by Charles V albeit at a high price. This started a lifelong rivalry between the Habsburgs and the Valois.

Henry VIII's marriage to Catherine of Aragon, daughter of Isabella and Ferdinand the Catholic Monarchs of Spain (or more precisely, or Castille-Leon and Aragon), was political. She was originally betrothed to Prince Arthur the older brother and heir to the English throne. When Arthur died, the 11-year old Henry was promised to her, partly so that the English won't have to return the dowry already partially paid. A papal dispensation from Julius II was obtained in support of Catherine's claim that her previous marriage was unconsummated.

When Henry ascended to the throne in 1509, he allied England with the Habsburgs, joining in the anti-French Holy League founded by Pope Julius II, and even invaded France alongside Spanish troops in both Aquitaine and Nord-Pas-Calais. Neither amounted to significant progress, so when Charles V's troops captured Francis I of France in the Battle of Pavia, he forced a treaty that didn't confer much benefit to England. Henry VIII was unhappy at the outcome.

At the same time, as mentioned, Henry VIII was very concerned with the security of succession in England. He started to have affairs with the Boleyn sisters, and was infatuated with Anne Boleyn. As mentioned, we know that up to the early 1520s he was still quite the devout Catholic, having a good relation with Pope Leo X who named him "Defender of the Faith."

However, the next few popes made relations with England complicated. Adrian VI, also known as Adrian of Utrecht, had been tutor to the young Charles V and even administered Spain in Charles V's absence. Leo X was a short-lived compromise pope. Clement VII was Charles V's chosen candidate, even if his reign was a turbulent period in Italy due to Francis I's invasion of Milan and northern Italy.

When Henry tried to obtain an annulment from Clement VII in 1527, it was at a very complicated time. Charles V's commanders in Rome had lost control of unpaid troops, who then looted Rome. Clement VII was imprisoned for several months. Regardless, Clement VII refused Henry VIII's two main points of argument: (1) a childless marriage was sign that it was blighted by God, and (2) his marriage had required dispensation in the first place, so therefore arguments could be made it was not valid.

A key change in Henry VIII's favor was installment of Thomas Cranmer, a close friend of the Boleyn family, as Archbishop of Canterbury. Henry VIII made it such that Cranmer was consecrated without taking an oath of loyalty to the Pope; thus, Cranmer could immediately grant the annulment Henry VIII sought. This set the stage for the "Reform from the top" that was mentioned.

Part 2.

Now we have in 1533 England Henry VIII, with Cranmer the Archbishop of Canterbury. In theology, the Church of England of 1533 is very similar to that of the Roman Catholic Church. Neither Cranmer nor Henry VIII pushed for any significant reform. Many bishops refused to accept Cranmer's authority. Henry and his allies then pushed for legislation that declares the royal head of state to be the head of the Church of England, which passed as the Acts of Supremacy starting in late 1534. This was strengthened by the Treasons Act, which was used to persecute Catholics such as Thomas More. This allowed them to dissolve monasteries and confiscate their property.

Skipping ahead, Henry VIII did get what he wanted: a divorce (which turned out to be several), but not yet a male heir, and the succession of that male heir to the throne. Through this era were developments of the idea of Protentantism in continental Europe, with major events such as Martin Luther's rise, the formation of the Schmalkaldic league in Germany, iconoclasm in Germany and the low Countries, the Confession of Augsburg, and many others. As Henry VIII had briefly married his fourth wife Anne of Cleves, he had to pay attention to the politics of Germany at the time of the Schmalkaldic league. This exposed Henry VIII to the theology developed by Luther and Melanchthon.

When Edward VI succeeded Henry VIII, many changes were pushed forward by the reformers, one of the tools of propagation being the Book of Common Prayer. Compulsory clerical celibacy among others, while maintained under Henry VIII, was finally overturned along with many doctrines. The new doctrines were then codified in a reform programme.

Unfortunately for Cranmer, and possibly for the Church of England, Edward VI was succeeded by Mary I who was a strict doctrinal Roman Catholic. Reformers of the Church of England were tried and condemned to death.

It is informative to look at the rest of Europe at this time. Mary I was the only child of Henry VIII and Catherine of Aragon, and she was born in 1516. Edward VI, her younger half brother, was born in 1537. In 1553, he died 16-year young, whereas she ascended as a 37-year old woman. Clearly, finding a husband and producing an heir was a very important concern, but she had to first find a suitable partner. Her cousin Charles V suggested that she marry his 26-year old son Philip II, who at the time was a widower but had one son from the first marriage. After negotiations, the marriage was celebrated in summer of 1554. Both Philip and Mary were staunchly conservative Catholics.

At this time, Charles V had started to abdicate his various titles, either to Philip II his son or to his brother Ferdinand, who became Holy Roman Emperor. In the years prior, he had made himself busy designing how his vast empire would be administered. He strongly wanted Philip to be King of Spain, and he offered Philip to consider which of the other possessions he things he wanted to rule (an exception being Austria and many HRE lands who were already quite firmly in Ferdinand's control). The Low Countries was very important to Charles: he was born there and he grew up there. Philip decided that he wanted to rule the Low Countries. This meant he also needed control of either the sea lanes between Spain and the Low Countries, or the land routes from Italy to the Low Countries, preferably both.

The orbit of Spain - England - Low Countries was a very important factor for Mary I, Charles V, and Philip II.

We know now that after Mary I's false pregnancy, she did not produce an heir and the throne went to Elizabeth, a Protestant. But Elizabeth's England and Philip's Spain did not immediately became enemies. And in fact, some in England suggested that Elizabeth marry Philip. For the first ten years relationships were cordial, even if they were moving in opposite directions.

The successes of the Duke of Alba in fighting against the Dutch rebellion forced Elizabeth to increase her support of the rebels. The presence of a large Spanish force just across the narrow ocean worried English reformers, whose memory of Mary's persecution was fresh. The relationship between England and Spain was significantly hurt when England seized Spanish ships carrying bullion that were meant to pay for Alba's troops.

Even if relations were very sour for the next few years, neither side escalated the tension. Elizabeth's primary threat at that time was Mary Queen of Scots, who was Catholic, and was strongly aligned with the Valois of France. So Elizabeth wanted a cordial relationship with Spain as a counter-weight against France. Unfortunately for Mary Queen of Scots, this meant that she had the support of neither Spain or the Vatican, as both were aligned together against France.

Years later, this enmity led to the Spanish Armada's attempt to rendezvous with Spanish soldiers in the low countries to carry them to England as an invasion force.

If we are speaking of Tudor England, it's religion does not change quickly. However, to answer this question fully a broadening of perspective is required beyond monarchs and nobles.

What do we really mean by a region having a religion? Is modern day England an Anglican country, even though most of its population is not practicing? It is important to appreciate that a ruler or tradition may make a religion established, yet this does not mean that a country truly professes that religion.

To be more direct, it is illuminating to contrast the English and Scottish experiences of Reformation. In Scotland, the Reformation begins later, 1560 in earnest. The key difference here is that in Scotland, the Reformation comes from below, with ordinary people playing a significant part in propelling it forwards. This was bound up with high political concerns too. Resentment of the French presence in Scotland was growing, despite the long ties between the two countries and reform provided a platform for nobles to launch their opposition. Popular and aristocratic participation ensures that, after the Scottish Reformation, Lowland Scotland is definitively Protestant. Their national church, the Kirk, reflects the more popular basis of their Reformation, operating without bishops or a strict ecclesiastical hierarchy. A final note worth considering here is that the Highlands are largely insulated from this and much of the region remains Catholic.

Turning to England, we can see that it is clearly reform from above, by the monarchy, for the monarchy. When we consider its wider approval, we find the English people far less receptive. Indicative of this is England's string of sixteenth century rebellions. Henry VIII deals with a number of rebellions in the North after his break from Rome, collectively called the Pilgrimage of Grace. Similarly Edward VI experiences rebellion in the South West known as the Prayer Book Rebellion, a reaction against the imposition of the English language Book of Common Prayer. Complicating this pattern, we also have a Protestant uprising known as Wyatt's Rebellion against Mary I. Elizabeth also finds herself facing insurrection. The Rising of the Northern Earls, rooted in a plot to install the Catholic Mary Queen of Scots on the throne, which at least initially raises a substantial number of people.

The truth is, English rulers changed the established religion, but most of England's transition to Protestantism was more gradual and many resisted it. Throughout the sixteenth century, Catholics and Protestants are both present in large numbers with a significant variation between regions. This means we can look at the persecution of Protestants under Mary I and Catholics under Elizabeth I as possible because there was a large enough population of both denominations, not because one or the other had become wildly popular.