The likely heir would have been James Francis Edward Stuart. His father was the last Catholic king of the United Kingdom, King James II in England and Ireland and known as James VII in Scotland. He fathered a son, James Francis, who was baptized and raised as a Catholic. This birth sparked fears of the nobles, as England had been Protestant for nearly 150 years by this point and was still deep in its rivalry with Catholic France. James was removed by the Glorious Revolution of 1688 but still tried to reclaim his crowns from exile, leading a serious Jacobite rebellion in 1690. He was succeeded in the throne by his son and law/nephew and his daughter, the dynamic duo of William and Mary. These two died without children, Mary in 1694 and William in 1702. In 1701, afraid of the crown becoming Catholic again, Parliament passed the Act of Settlement, declaring that only Protestant descendants of Sophia of Hanover could become the monarch of the UK. After William's death, James II/VII's second daughter, Anne, became Queen for the next 12 years until she also died without children. This is when the Act of Settlement kicked in, making Sophia's son George I the king, the first Hanover king of the UK. However, James II/VII took over from his (legitimate) childless brother Charles II and Anne took over from her childless sister Mary, so James Francis Edward Stuart should have taken over after William's death in 1702 and become James III/VIII because he was Mary's oldest brother and William had no siblings. So he also should have taken over after Anne's death 12 years later. And yes, baring any revolutions, assassinations, or other unforeseen events, the Catholic descendants would still be the monarchs, although the House of Stuart has been extinct since 1807. There is still a Catholic claimant to the British throne, but it gets really messy to trace a theoretical claim through 300 years of intermarried nobility.

At the time, it would have been Anne's half brother James (the Old Pretender). He had two sons, Charles (the Young Pretender) and Henry. Both died without legitimate children, however. Of course without the Glorious Revolution, James and his sons would have been kings before Mary and Anne.

Assuming both Charles and Henry still died childless, the succession would have likely followed at least some of the Jacobite line. This being all speculation, of course, but the nearest cousin I can find is Charles Emmanuel III of Savoy/Sardinia and his offspring. This Charles was the son of Anne Marie d'Orleans, herself the daughter of Princess Henrietta Stuart of England, Duchess of Orleans, the sister of Charles II and James II. If we follow the exact Jacobite succession, the current ruler would be Francis II (aka Franz, Duke of Bavaria). I don't know if these proposed Sardinian kings would have done what the Hanoverians did and have the senior heir take both titles--that's quite the geographic stretch--or if it would have been split between available sons. I would have to research each person in the Jacobite line to see if the claim followed the senior heir in every generation. (Willing to do so; just a little short on time at present)