Currently the vast majority of Catholics in England support the current Queen Elizabeth II, even if a Catholic believed Elizabeth I was illegitimate that wouldn't effect the claim of Elizabeth II. Historically Catholics who opposed the monarchy were Jacobites. Jacobites believed that King James II remained the King of England even after the "Glorious Revolution" of 1688. James II was born a protestant but later converted to Catholicism in 1668.

The eldest daughters of James II were raised as protestant on the orders of their uncle Charles II. After Charles II died in 1685 James II became the King of England. Protestants were willing to abide having a Catholic King since his daughters were protestant, and after James II died England would have a protestant monarch again.

However in 1688 James II had a son, James Francis Edward Stuart (also known as the old pretender), this caused a group of Protestant nobles to overthrow James II and place his daughter Mary II and her husband William III on the throne.

The Jacobites continued to claim the throne with little success until Anne I, the last Stuart monarch died in 1715. Afterwards the King of Great Britain would be George I who was distantly related to the royal family and who only became King due to the Act of Settlement of 1701, which among other things banned Catholics from inheriting the throne.

The Jacobites had a number of uprisings in the 18th century to reclaim their throne. Most notable the risings of 1715 and 1745. However after the death of the Old Pretender, and his eldest son the Young Pretender, Jacobite risings in Britain ceased. And after the death of the Cardinal Duke of York in 1807, who had no children because Catholic Cardinals aren't allowed to marry there were no more legitimate descendants of James II. The Jacobites stopped having significant support.

According to Jacobites the descendants of James II youngest sister Henrietta Anne are the legitimate claimants to the English throne. The Jacobite claim then passed through the Houses of Savoy, Austria Este, and Wittelsbach. The current Jacobite claimant is Franz, Duke of Bavaria.

Although if we look at the alternate line of succession from the death of Mary I in 1558 we get a different result. If Elizabeth was banned from inheriting the throne. Then according to the Third Act of Succession the descendants of the youngest sister of Henry VIII would be next in line to inherit the throne, if all of his children died without issue. Therefore the only remaining claimant would be the descendants of Mary Tudor, Queen of France.

Lady Jane Grey who was the granddaughter of Mary Queen of France briefly became Queen for 9 days between the death of Edward VI, and the succession of Mary I. This was an attempt to prevent the Catholic Mary I becoming the Queen of England. So if Elizabeth was excluded the heir to the throne would be the mother of Lady Jane Grey. Francis Grey the Duchess of Suffolk. This would have resulted in a Queen who was more protestant than Elizabeth. The current heir to Francis Grey would be Lady Caroline Villiers.