Molotov

• Commonwealth of England (Serenisima): Historic England, Jersey
• Kingdom of Scotland (Oyid?): Scotland, Isle of Mann
• Republic of Ireland (Carpe): Ireland

The British Isles have diverged greatly due to one critical event: The fervent nature of the Diggers. Spurred on by what they saw as Cromwell trying to import the Venetian oligarchy and take away the chance of a parlimentary, democratic structure in favor of the dictatorship of the few, popular unrest proved to be too much to contain: The power structures fractured, with Cromwell being unable to maintain the support of his New Model Army when so many of the men resented their orders in battle after battle,and the Scottish and Irish quickly taking advantage to throw off what was seen as an oppressive, Protestant English Yoke.

When the dust settled,the Digger's militias, swelled by the ranks of defectors and veterans who had settled their desire for peace, emerged as the ruling entity on English clay. The Scottish had reasserted their own kingdom with Charles II as King (as well as with the newly acquired Isle of Mann), and the Irish with a local king of their own. The colonial acquisitions were, in the end, handed over to the Irish and Scottish crowns for safe-keeping, the Diggers finding that stealing the Native's land when they had plenty of their own to farm to be a bit against their tastes.

Since then, little has been done against the long-standing order within the British Isles, the Scottish Crown and Diggers having proven themselves reliable and stable for their peoples, with the only trouble being that the Irish Crown gave way to the Republic--and quickly settling back into its mold. The long prosperity having profited all three nations. Industrialization swept over them, and each in turn saw the erection of local factories for their goods, each becoming a thriving group on their small island home, participating in the trade but mostly being removed from any international conflict on their shores, instead participating in domestic wealth generation, colonial experiments, or global commerce.

• Faroe Islands: Denmark?
• Iceland: Norway?
• Shetlands: Scotland probably

• If we take it as a given that England, Scotland, Wales and Ireland are ruled by Cromwell until the time of the Rump Parliament being dissolved (1653, the year Barebone's Parliament is established, could be the trigger for the Third English Civil War), that makes things a little simpler. Cromwell had already fought factions of the New Model Army; we can say it all goes to shit in 53-54, around the time the Anglo-Dutch War is historically finishing up.
• Scotland, by this point, is still reeling from the Cromwellian Invasion; the Covenanters are broken, and Campbell power lessened. Glencairn's Rising is the most able to benefit from chaos among the English forces; and despite divisions between Highland troops and Lowland officers, Glencairn remains commander of the force, Middleton and Munro becoming deputies despite their tensions. In 1654 the Battle of Dalnaspittal is in reverse from history: Monck is defeated, his forces smashed, and the following years see a Highland army join Lowland risings to expel the English armies by 1655, though low-level rebellions continue. Nonetheless, Charles II becomes a reluctantly restored King of Scotland.
• I'd imagine most of the proceeding relations with England would be pretty bumpy, as royalists of all stripes, covenanters, Highland chiefs, and religious radicals oppose each other, with some clamouring to invade England. Quite possibly there would be an aborted invasion attempt, with an eventual peace deal that satisfies honour on both sides but leaves the Stewarts unable to make a jump at the restoration. Depends on how Seren sees the English Commonwealth developing.
• Regarding domestic development, the polity would be less Anglicised than IRL; but still Lowland and English exile dominated. The church would veer between Episcopalian, Presbyterian and Catholic risings; potentially this contributes to general instability until freedom of worship is granted, following the examples of English development. But the Scots would be wealthier and freer to take a leading role in the north Atlantic, perhaps involving themselves from Shetland to Iceland; and royalist support would mean they'd inherit a navy to control the colonies.
• The Stewarts would realistically try to improve ties with Ireland after Charles' death; we can play with the idea of royal marriages etc leading to a more Gaelicised country after Charles II's death, especially once the old royalists die off and the military power of the clan system is allowed to flourish, even as commercialisation goes ahead. Into the 19th century an active cultural Gaelic revival - think Ossian on steroids and in actual Gaelic - in both countries might contribute to increased ties. The Irish Revolution probably leaves the Scots reeling, and attempted revolts among some of the Lowland Presbyterians might be likely, but a legacy of religious tolerance and the loyalty of the clans to the principle of monarchy would keep the regime intact even as all the rest of the British Isles goes republican. Scots would still be spoken, but quite possibly as a minority language in the Borders, Lothian, and the very east coast. Perhaps there's even a Gaelic dynasty on the throne, given the chaos of the revolutions, and the march of time for the House of Stewart
• Economically, the role of the North American and Caribbean colonies, the dominance of Scotland's fleet in the North Atlantic, and the lack of a common market with England and Ireland would have a striking effect. With no cultural chauvinism/racialism + the 45 impeding development, the Lowlands and the Highlands would likely develop on something of a more equal footing; the uplands being given over to cattle and sheep without much clearance, and with more fishing, factories etc being built in towns in the north and south. That being said, depending on when the industrial revolution happens, the influence of England could also be seen in more retention of communally worked lands in both the Highlands and Lowlands.
• The population will probably be higher than IRL, if the Highlands are able to develop properly, giving more of an impetus for colonisation; Gaelic diaspora in the North Atlantic as well as the colonies? Fascism, ultra-monarchism - since the royals are still around, albeit not as powerful in a parliamentary democracy - and a residual Highland-Lowland conflict could all play into problems for Scotland, especially if other countries seek to dislodge her control of trade and fishing in the North Atlantic. Have they done anything with Svalbard/Iceland/Faroes/Greenland?