The problem is, the Stuarts would have to be satisfied with merely Ireland, with giving up claim to England and Scotland. I mean, sure, England and Scotland go to William and Mary (and then Anne and then George), Ireland stays with James (although it's probable that the Ulster Scots leave, either back to Scotland or on to America), it's perfectly rational, but that's not how royalty works. It's the same attitude that kept the Hundred Years' War going for so long, albeit that attitude is fraying a bit. During the OTL Jacobite Risings, particularly the '45, one of the problems was the Stuart being in conflict with his, largely Highland, Scottish allies over taking the offensive. They believed that if they aided Bonnie Prince Charles in retaking England, they'd be doubly marginalized, because that was their status before. Mind, things weren't really better for them under Hanover, and that's partially because of the Risings. Anyway...
A Stuart-ruled, independent Ireland would be very close to France, toeing that line carefully. Obviously there would be oodles of butterflies, so that the US is likely to not form, or to form far differently from OTL, likewise Canada, New Zealand and so on.
One of the big changes would be that Ireland would take part at least peripherally in the proto-Industrial Revolution of the 1700s, probably becoming an important textile-manufacturing country, and possibly getting into shipbuilding. With shipbuilding could come colonies but where? Newfoundland might be possible, I believe its strategic value had diminished by this time and both France and England/Britain (I assume we'd still get the Act of Union 1707 to prevent the possibility of Scotland pursuing an independent policy) might appreciate having a buffer in the area. Possibly a minor Caribbean island, a couple of ports in Africa and India, but it's important to remember that Ireland is still not highly populated, so colonies would be limited.
However, with industry as an important sector of the economy, and not focused in Dublin and northeast Ulster as OTL, and with land laws that favor Catholics instead of the opposite, there wouldn't be an enormous potato-dependent and vulnerable population to die and emigrate when Blight comes. I think that's important to understand, in a five year period it's estimated that a million people died from starvation and disease and so on, while another million emigrated to Britain, North America or Australasia, continuing to pour out for the next century. As I recall, Ireland's population only start to gain again in the 1960s, and the whole island still isn't as populous as it was on the cusp of the Famine (about six and a half million in both parts today, as opposed to eight million in 1840). Ireland's diaspora is huge, and it's entirely because of that. Without it, Irish emigration would look more like Italian or Polish emigration, with many people intending to go overseas, make a buck and come back home having moved up a couple of stations. Irish immigration to the US was unusual in that most who came stayed, whereas the usual pattern was for about half of immigrants to stay for less than five years, work what job was available, live in poor conditions and save a pile of money in order to buy land or start a business back in Italy or so on. For us in the Irish diaspora, that wasn't an option. Ireland was dead and going back wasn't in the cards.
Here though, no Famine is a given. Catholic land-ownership would be much much greater, with Protestants being dispossessed, possibly even expelled (and certainly encouraged to move on). So, far fewer uneconomically tiny plots of land on which can only be grown potatoes. More grain, fruits and stock raising. More industry. More trade and some few colonies and settlement possibilities in Newfoundland. Think the Netherlands or Denmark, since Ireland doesn't have the coal to pull a Belgium.
Now, back to the main point of your question (now that I've rambled on for so long, my apologies), I would expect that the main focus of this smaller Irish emigration would be France and Spain and their colonies, assuming they allow it. Spain in particular was picky about who settled where, with Aragonese subjects not allowed to settle in Castilian colonies for the longest time and I'm not sure when that actually changed. OTL though, the Irish Wild Geese and descendants played an important part in Latin America, before and after independence. I see no reason why that would be different here, with northern Mexico (OTL California-Texas), Argentina and Canada (OTL Quebec-Ontario) at least having important Irish populations.
Actually, if it starts early enough, Irish immigration could really shore up Canada's population. They'll never match the British colonies to the south, but if the Irish are assimilated, it would maintain the French character of the colony in its whole, instead of half being anglicized. Oh, one more thing, Britain's population would be a fair bit lower, and France's and Spain's just a touch higher.