Was it possible for Catholics to express their faith openly (if at all) in Britain during the time of the of English persecutions (April 29, 1559)-(April 13, 1829)?
On 29 April 1559, the English House of Lords by 33 votes to 12 passed a bill abolishing papal supremacy over the Christian church in England, and establishing the supremacy of the English monarchs over it. Also in April 1559, a bill abolishing the Mass and imposing an English language Book of Common Prayer liturgy passed in the House of Lords by a majority of three and was implemented on 24 June of that year. To refuse to take an oath of belief in royal supremacy over the church became a crime punishable by removal from public office and inability to hold any office. To defend papal authority over the church became punishable in the first offense by loss of goods; the second by imprisonment for life; the third offence was considered treason punishable by death. - The Persecution of Catholics in Elizabethan England
The persecution against Catholics officially ended on April 13, 1829.
The Roman Catholic Relief Act 1829, passed by Parliament in 1829, was the culmination of the process of Catholic Emancipation throughout the UK. In Ireland it repealed the Test Act 1672 and the remaining Penal Laws which had been in force since the passing of the Disenfranchising Act of the Irish Parliament of 1728. Its passage followed a vigorous campaign that threatened insurrection led by Irish lawyer Daniel O'Connell. The British leaders, starting with the Prime Minister the Duke of Wellington and his top aide Robert Peel, although personally opposed, gave in to avoid civil strife. Ireland was quiet after the passage.
The Act permitted members of the Catholic Church to sit in the parliament at Westminster. O'Connell had won a seat in a by-election for Clare in 1828 against an Anglican. Under the then extant penal law, O'Connell as a Roman Catholic, was forbidden to take his seat in Parliament. Sir Robert Peel, the Home Secretary, until then was called "Orange Peel" because he always supported the Orange (anti-Catholic) position. Peel now concluded: "though emancipation was a great danger, civil strife was a greater danger." Fearing a revolution in Ireland, Peel drew up the Catholic Relief Bill and guided it through the House of Commons. To overcome the vehement opposition of both the House of Lords and King George IV, the Duke of Wellington worked tirelessly to ensure passage in the House of Lords, and threatened to resign as Prime Minister if the King did not give Royal Assent. - Roman Catholic Relief Act 1829 (Wikipedia)