As far as I know, the reason that the English church broke from the main body of the Catholic church was because Henry VIII wanted a divorce and the Pope wouldn't give him one, i.e. a politics-driven decision rather than a belief-driven decision. However anti-Catholic prejudice eventually grew and their rights were restricted until the 19th century. When, why, and how did this change come about?
Definitely between Elizabethan era and Stuart dynasty. By the time Elizabeth took the throne, England was only recently turned Protestant (although there had been a see-sawing of Catholicism and Protestant due to Queen Mary, who wanted to re-Catholicize England). Elizabeth was more or less lenient to the Catholics until the release of the Papal bull Regnans in Excelsis that essentially excommunicated Elizabeth.
While this may not have been a big deal to the Protestants, the massive Spanish armada gathering in Lisbon to invade England was no joking matter. England and Spain had been at odd with one another for some time due to Elizabeth's support for privateers who were raiding the Spanish treasure fleet from the New World. With the Pope excommunicating Elizabeth as well as Catholic forces, blessed by the Pope planning an invasion of England and a very popular Queen, the public quickly turned against traitorous Catholics in England. Combined with several Catholic plots (Throckmorton plot, Babington plot) to assassinate Queen Elizabeth really forced all Catholics in England in to hiding.
Perhaps the most famous case of Catholic plot is the Gunpowder plot in which Guy Fawkes tried to blow up Parliament and kill King James, who ironically was probably a secret Catholic himself(or at the very least tolerant to Catholicism). The intensity of distrust and hatred of Catholics should be noted here as this event gave immense popularity and support to King James VI and I, which he used to get more money from the Parliament.
Source: A Monarchy Transformed Britain 1603-1714