The short answer is no, Catholicism did not completely disappear in England at any point, though it was certainly suppressed to varying degrees at different times, before its eventual legalisation after 1829. Particularly in places that were more distant from central authority, such as Wales and the north west, Catholicism continued to be practiced throughout this period.

In fact, after Henry and Edward's reformations, England would have two openly Catholic monarchs, and two more who were commonly suspected of 'cryptocatholicism' (and not entirely without cause).

When Mary I took power in 1553, she attempted to reverse the reformation undertaken by her predecessors, which involved a bloody campaign of suppression against prominent Protestants, which resulted in several hundred Protestants being executed by burning.

This experience left a lasting impact on the psyche of English Protestantism, and for the next few hundred years, fear of Catholic influence would lie at the heart of many of the key events of English history, such as the Civil Wars and Revolution of 1688. Even the American Revolution had an anti-Catholic element to it- one of the colonists' grievances was the fact that the British Parliament had effectively legalised Catholicism in Quebec.

So when Protestants retook control of England following the coronation of Elizabeth I, various measures were introduced to suppress Catholicism. For ordinary people these mostly took the form of laws that threatened fines or imprisonment for things like failing to attending Protestant communion, and barring Catholics from holding certain public offices. Actual priests, however, were viewed with great suspicion and thought to be plotting a return to the days of 'bloody Mary', and they were treated extremely harshly, to the extent of torture and execution, particularly for those known or suspected to be Jesuits. Essentially, the focus was less on stamping out Catholicism altogether, than in suppressing its political impact. The great fear was that Catholics were loyal to the pope rather than the king or state, in a similar manner to how anti-Semitic tropes in the 19th and 20th centuries portrayed Jews as having international loyalties which trumped their national ones.

Following Elizabeth, however, the Stuart dynasty took power, and they had a complicated relationship with Catholicism. Though they were nominally Protestant, many English Protestants felt they had pro-Catholic sympathies, and they were not entirely wrong- they reoriented English foreign policy to be more pro-Spain and France, and their theologians tended to favour more 'Catholic-style' forms of worship than some Protestants approved of. Charles II is thought to have converted to Catholicism on his death bed, and James II was an openly practicing Catholic, although this is part of the reason he was overthrown in 1688.

This is a simplification, but its helpful in this period essentially to think of Protestantism and Catholicism in England not as binary opposites but as a spectrum of sorts. Puritans and Presbyterians often wanted to expunge any trace of 'Catholic' worship at all, whereas 'High Anglicans' favoured retaining a lot of Catholic practice in worship, while rejecting the authority of the Pope. The Stuarts, and many of their supporters, were on the 'more Catholic' end of the spectrum, even when they were still Protestants. Many of the 'hotter' (less Catholic) Protestants did not approve of this, and actively pushed back against the Stuarts. The Civil Wars were caused by many factors, but anti-Catholicism certainly played a part. There was also massive resistance to the idea of allowing James II to succeed to the throne at all, and when he got the throne, he only lasted 3 years before he was overthrown.

If I get a chance I shall revisit this later and add more about the role of Catholicism in the 17th century, but the essential point is that Catholicism did not entirely disappear either among the common people or elites, and played an ongoing role in politics throughout the period between the reformation and emancipation of Catholics after 1829.