If Cromwell's Commonwealth continued, under a more talented leader than his son, what is the most likely end state for the British Isles? To me, the state power, the religious enforcement and the ethnic cleansing point to a more brutal, less pleasant history for the islands. But they also point to a place that is more religiously unified and also more ethnically unified.
The Celtic Irish will be effectively pushed to hell, Connaught or Jamaica. The Highlanders will be "othered" and likely wiped out, so unlikely to be reincorporate into Scottish identity in the 1800s. And both Ireland and the Highlands will likely have more plantations from Lowland Scots/English stock. The English, Anglo-Irish and lowland Scots will likely all end up with a common Calvinist religious faith, weakening the difference between them. And likely they will all have a single legal system and unitary governance.
The combination of this points to an eventual unified British Isles identity, despite the name of the Commonwealth specifying England, Ireland and Scotland separately. I suspect the name would change in time, and people would consider themselves primarily British, just as the French Revolution and its resulting political structures ended up wiping out Breton, Norman and Provencal as primary identities.