Your POD could be that the invasion happens when it was supposed to, on January 20 instead of March 7. The extra month delay only helped the British; it just gave them time to prepare for the French invasion.
The French had twelve thousand men commanded by Maurice de Saxe, and a French estimate in 1743 put the British at sixteen thousand men - but that army was scattered all over the country, with garrisons all over Great Britain. If Maurice's army had invaded the month early, there's a good chance he would have faced and crushed a hastily gathered smaller British army commanded by George Wade (something between 9-10,000 men) somewhere between London and Maldon. And, even if the delay still happens, there's a good chance that Maurice de Saxe would defeat the British, since here they would have the superior troops.
From there, that'd probably be the end of the war. The British would surrender, and the Hannoverians would have to be pushed out in favor of the Jacobites. The war with Austria was in a bit of a lull period, so the Second Silesian War doesn't happen, and the House of Wittelsbach probably maintains control of the Holy Roman Empire, even after Charles Albert of Bavaria dies the next year. Bavaria, which still holds Bohemia, and also has the HRE, would be a significant third power in Germany.
As for a Jacobite Britain... initially, it'd be tough to maintain, for obvious reasons. If the French garrison Britain, that's only bound to cause more insurrection; if they don't (which I think is more likely, because it is the better route, albeit still risky), then there's less to protect newly crowned James III. And James is unlikely to make concessions to his faith... so that doesn't help the situation. "Bonnie Prince Charlie", however, is fairly likely to convert to ensure being King. There's a chance that measures might be taken to "speed up" the succession process after a particularly nasty revolt - either by forced abdication or assassination - to get Charles on the throne as a good compromise candidate. A reign of a protestant Charles III would bring some stability back to a country that really needs it; under him, some sense of permanence can be established for the Stuarts and for the Kingdom. If not... there'll be turmoil and bad blood brewing in Britain for quite a while, because James III lived quite a while longer IOTL (died IOTL in 1766 at the age of 78), and it might just get worse after his death.
I don't think it's very likely that the Hanoverians manage to reclaim their crown. Even if something as bad as a Civil War came up during James' reign to get George II on the throne, the rebels would be unlikely to get George II back from Hanover in a world where the French had just recently invaded Great Britain - too much of a (probably) limited royal navy would be loyal to James, plus the French would be watching for it, and would put forth full efforts to stop such a plan from working.
In any case, a Jacobite Britain is a lot more likely to be friendly with France (under James III, almost like a vassal; under Charles III, an ally), which gives France a lot more room to work with on the continent - there's really no one state that can oppose French Hegemony. In response, there's a chance that coalitions are formed on the continent against French Hegemony - but that can only work if Frederick and Maria Theresa are willing to work together... plus the House of Wittelsbach (i.e. the HRE) would be pretty friendly with France... so, actually, at least initially, you might see more carving up of the Habsburg Empire, before you see any coalitions against French Hegemony.
Abroad, with a Jacobite Great Britain, France would probably get India, and might get some concessions from the British in North America.