I think it would make the colonies more likely to stay with Britain. The Commonwealth would become much more Calvinist, which would be a better fit with New England. I can't see the slave states breaking away on their own.
Not more ''Calvinist'' but more open to religious tolerance within certain limits - that was the Army's and Cromwell's official position on the issue - and Puritan. For example they would not accept the old Church of England nor Catholics.
The existing divide between the mercantile and craft oriented New England and the plantation economy of the South would remain in place.
The issue will be the Catholics in Maryland (and perhaps Virginia), were a civil war was fought in OTL between royalists and parliamentarians, with the later winning it and establishing a commonwealth. Another outcome of things is what happened with Bacon's Rebellion (Virginia).
There is at least a possibility that representative democracy takes place earlier in America than in OTL. Assuming that the successors of Cromwell accept the Leveller ideology of John Evellyn and others (which had some support in the Army) then something akin to anti-class-conscious democracy becomes an acceptable model for government. Oliver Cromwell surpressed such ideas during his rule but the ideas were out in the public. The Puritans were more likely as a group to be of the lower or " middling" classes and suspicious of the hieirachical structures of the Anglican Church and the nobility. With no King there won't be Royal Governors and the colonial governments might be organized as Commonwealths.
I think it is unrealistic that the Levellers and similar groups gain general acceptance. They were repeatedly suppressed and purge from the Army, Parliament, etc.. Their opinions were in minority in the City and among the gentry. Parliament, Cromwell, the Grandees, the Army and Gentry were conservative in matters of property and political representation.
The incipient republicanism would lead to more commonwealth styled governments in the North American colonies. That is to say representation of the colonist in the legislatures and executive, with an elected Governors or named by their own executive bodies, given de facto in self rule.
The downside of all this is that the Indians (New England and South) would be victims of the land and expanding colonization of the colonies. Slavery (South) will become part of the society and plantation economy.
The Acts of Navigation and similar legislation would give London legislative powers in trade and taxing. Nobility will perhaps be open to the growing group of merchants or given out as political sinecures or rewards of political services.