The simple answer is that it is still possible to find persecution going on of both Catholics and Protestants.
It was only in 2013 that is was decreed that a member of the Royal family in line to the throne could marry a Catholic.
It is not British persecution, but an English one. This started with the action commenced by dear old Henry VIII and his Reformation of the English and Irish churches and his Act of Supremacy of 1534 that ruled the reigning English monarch was the supreme head of the Church of England. This Act was repealed in 1554 by Queen Mary I, a very devout Catholic, but due to her record whilst on the throne, and the threats made by the Catholic Church towards Queen Elizabeth I, the Second Act of Supremacy was decreed in 1559. This reinforced the legal fact that Queen Elizabeth I, and all subsequent English monarchs, would also be the Supreme head of the English Church. Any allegiance to the Catholic Pope and his Church would thereafter be considered treasonable under Henry VIII's Treasons Act, the first victim of which was Sir Thomas More who fell foul of dear Henry by refusing to condon the King's actions. The Spanish Armada of 1588 that was launched by the Catholic King Phillip II, heavily supported and instructed by the Pope, to remove the Protestant whore (Elizabeth I) from the throne and make England Catholic once more, stirred up very great anger and suspicion by the English of all Catholics, which continued for at least 300 years.
Because of that history, during and since those times there has been constant mis-trust of Catholics in the eyes of the Protestants, and visa versa, depending who was on the throne. Various Acts of terrorism, most famously the Gunpowder Plot of 1605 and even the Great Fire of London of 1666, did not help the Catholic cause with them being held responsible for both, correct of course with the former, but unfair in the case of the latter. The 1689 Glorious Revolution saw the over throw of the very unpopular James II who had converted to the Catholic faith shortly before he was crowned, with the accession of William of Orange, King William III, a Protestant, to the English throne.
From that time on various Acts of Parliament tried to calm the Protestant v. Catholic issue down but the Jacobite Rebellions of 1715 and 1745 did not help matters, even with an Act of Union with Scotland of 1706 and 1707. The "Irish question" was also constantly causing friction, and during the 19th century The Great Famine in Ireland sending many Catholic peasants to England. Then the troubles across Ireland during the 20th century further fueled the antagonistic levels between the followers of the two faiths, with politics, as always being the case, driving the Protestant v. Catholic question onwards, with of course during the 20th century the added cause of a "United Ireland" taking the distrust ever further.
However, after that very brief summary that skims across much, including the thousands condemned to horrible deaths because of what they believed or who hated them the most, modern anti-Catholic feelings have greatly subdued across Great Britain since the Second World War, but still "the differences" continue, without now the pure acts of hatred to one another. As stated, even the Royal Family relented in 2013 on the question of Catholics entering their fold!