This would depend to a large extent on when, and how, the Jacobites would have succeeded. They could've (in theory) won any of the following ways:
- 1690 - James II/VII wins the Williamite War in Ireland and then mounts a successful invasion of Great Britain with French support. This would've required at least some Englishmen and Scots to get over their terror of being conquered by an army dominated by Irish Catholics.
- 1715 - The Hanoverian/Whig government fails to thwart the main components of the Fifteen before they even get off the ground, and the Old Pretender successfully takes the throne with French support. Certainly at this point the Tories were still a serious political force, and the Hanoverians weren't all that entrenched in power. That said, James "III" had such an uninspiring personality that he had a tendency to tank the morale of any army that had the misfortune of meeting the person they were fighting for.
- 1719 - This went more or less the same way as the Fifteen, but was even less well-organised, and had Spanish, not French, support. Including mostly for the what-if of someone other than France sponsoring the Jacobite comeback. (I shan't include the 'Swedish plot' of 1717, when the Jacobites gave Sweden a huge sum of money on the promise of an invasion that, er, never happened.)
- 1721/22 - If the Jacobites had even the slightest semblance of competence and organisation, they'd have exploited the collapse of the South Sea Bubble, in which George I himself and virtually the entire Hanoverian elite were implicated, and challenged for the throne, probably without even needing foreign help. However, they screwed up royally and the leaders of the closest thing they had to a coherent plot, the 'Atterbury Plot', were arrested.
- 1745 - The Forty-Five actually succeeds, again with French support, and brings the Young Pretender to power, governing in the name of his father. This one has the added benefit of having a more inspiring figurehead than the charisma vacuum that was James "III".
It's probably worth saying that, whatever happened, the chances of Britain actually being re-Catholicised under Jacobite rule would've been close to zero. James II/VII's attempts to introduce religious liberty for Catholics (and to put them disproportionately in high-up offices) went a long way towards sparking the Glorious Revolution in the first place. James was a stupid and terrible king, but he probably wasn't stupid or terrible enough to try and actively re-Catholicise the country after taking back power.
Also, the Jacobites took most of their support in England from hardcore High Church Anglicans who were legitimist Tories, and in Scotland from Anglicans upset at how William and Mary made Presbyterianism the state religion up there. This entrenched further after 1714 when the Whigs became the sole party of government, and granted some liberty to Protestant Dissenters in England. While the Jacobite 'royal' family were themselves Catholics, their support base came almost entirely from Tories who wanted to restore the supremacy of High Church Anglicanism.
Quite what effect this would've had on colonial missionary programmes, in India or otherwise, is up for debate. It would be curious, however, for a Jacobite state to start furiously attempting to convert Indians to Catholicism.
The line of succession would not have been amended to ensure that all future monarchs were Catholic, partly because the Jacobites' Anglican base would've hated that, and partly because the whole point of Jacobitism was that messing with the line of succession was just not on.
As for the state of the three kingdoms, Ireland would most likely have been restored to the status it enjoyed under Charles II - still an English satellite, and still mostly ruled by Protestants, but nowhere near the rigid Protestant Ascendancy entrenched under the Williamites and Hanoverians. Perhaps Irish Catholics might have ended up in a slightly better position than that if the Jacobites had won in 1690. And as for whether the pattern of events that led to the Union with Ireland would've happened - who knows?
On the island of Great Britain, provided the Jacobites won after 1707, the Union would've almost certainly been dissolved. In 1715, the Earl of Mar actually raised his banner with the coat of arms of Scotland on one side and "NO UNION" on the other. No idea whether the Union would've happened if the Jacobites were restored earlier - the idea was repeatedly mooted under the Stuarts, and wouldn't have become associated with Williamite/Hanoverian rule. The Anglican Church would've been restored in Scotland.
In terms of European policy, certainly the Jacobites would've allied with their sponsor in whichever European war had prompted them to support the Jacobites in the first place. But no alliance lasts forever, and it's worth noting that
Hanoverian Britain was allied to France between 1716 and 1731 (and relations didn't seriously deteriorate until 1742), and in so doing successfully suppressed Jacobitism for 20 years. The idea that a Jacobite Britain/England, 50-100 years after the French (if it was them) sponsored their return to power, would've been more inspired to intervene to save the House of Bourbon is for the birds.
Tories, indeed, traditionally followed a Blue Water Policy that sought to stay out of European conflicts and focus on developing Britain/England's position around the world. But then, during the Walpole era,
this was the exact policy pursued by the Hanoverian Whigs, and when popular opposition to Walpole's foreign policy grew in the 1730s, Tories and Jacobites were more than happy to jump on that bandwagon. So while, in general, a Jacobite state might have pursued more of a Blue Water Policy, this is far from certain.
Further afield, there probably would've been a conflict in America (and India, &c.) between Britain/England and France at some point in the 18th century, but whether that would've happened at the same time as the Seven Years' War, and whether it would've had the same result, is up for debate. It's entirely possible that France could've held Quebec and Louisiana, and maintained significant holdings in India. Certainly the Americans had very little time for James II/VII (see the Boston Revolt of 1689), and I don't think the American Revolution would've been averted under the Jacobites (it may, however, have occurred under different circumstances, and possibly had a different outcome).
Probably the most significant effect would've been on the internal government of the British Isles. I've gone over religious policy, foreign/colonial policy, and the Union/Ireland. One other issue would be financial policy - the Whigs and Williamites/Hanoverians sparked the development of the City of London and the creation of the national debt, and Tories and Jacobites traditionally opposed that, so a Jacobite restoration may have stymied Britain's growth as a financial power (which may in turn have impacted on the development of the Industrial Revolution).
And finally, and probably above all, the matter of the English/British constitution. James II/VII was a tyrant who, like most of his fellow Stuarts, had next to no respect for Parliament or for any limits on his power as monarch. The Glorious Revolution entrenched the principle of Parliamentary supremacy over the monarch, and under the Williamites and Hanoverians the use of executive power switched from the monarch themselves, to ministers accountable to Parliament. The monarch was left dependent on Parliament for their income and even for their place as monarch.
A Jacobite restoration would've retrenched the idea of the King, who derives their power not from the will of Parliament but from pure legitimism, personally leading the government, with Parliament taking a much smaller role than it enjoyed under the Williamites and Hanoverians. Constitutional monarchy along British lines either wouldn't have developed, or would've looked totally different. And that's before we get into the issue of the constitutional structures of Scotland without the Union.
As the wall of text above demonstrates, this is a
massive counterfactual and I've probably only covered the tip of the iceberg in terms of potential implications. There's so many permutations in so many areas that it's nigh-on impossible to come up with credible answers in a lot of cases. But what I can say is that wholesale re-Catholicisation would not have happened, Ireland would not have become an equal partner in the union of the crowns, the line of succession would not have been changed to exclude Protestants, England/Britain would probably not have been any more inclined to stop the French Revolution (if it happened in this timeline), and the Jacobites wouldn't have tried to aggressively convert India to Catholicism.