Certainly Europeans probably wouldn't looked kindly on the Republic, although England wouldn't face the same immideate threats as France (depending on the navy, I know RN flourished under Cromwell but what effect the Levellers have I dunno.) but I can see France and Spain taking very dim views now that their old enemy has turned even more into a threat on their systems of government.
Beyond their basic beliefs and the Putney Debates, I have little knowledge of the Levellers. Although their democratic ideas were obviously radical, were they keen to spread the Revolution?[/quote]
My route for Leveller success to a POD where the pro-Leveller officers organize themselves better, radicalize the Heads of Proposal a bit, and thus accept the Agreement-Heads of compromise w/o attempting any mutinies. For my purposes have Charles I escape later would have these effects (though after doing some reading I'm beginning to think Cromwell let him escape in order to have an excuse to take out the Leveller leadership in the NMA).
Also how about their effect on the American colonies? Although not very well established by the 1650s, would anti-Leveller sentiment be high? Or would the Puritans and Nonconformists embrace it?
The Levellers were very pro-Independent, with the Agreement of the People having in it the right of congregations to choose their own minister. The Puritan preachers who were attached to the NMA's units were key in radicializing the regular troops to the point that they were willing to support the Agreement, so I think that the New England colonies (founded, settled, and ruled by Purtians) would accpet the Leveller government (you should see some Massachusetts Bay Colony learning-to-read kids books that explicitly reject the divine right of kings to rule, as in rule anything).
Virginia might be leary, but wouldn't really have a great deal of choice. Neither the Commonwealth or the New England colonies OTL seemed to have a problem with slavery, so I don't think that the tobacco farmers of Virginia will have any issues with the new government.
What role would the NMA have in this new Republic? How much influence would they bring to bare? Will the term "Roast Beef Republic" come to stand for corrupt, unstable government?
The Levellers appeared to want some level of demobilization, although that desire is going to be tempered by the need to fight off Charles II's invasion.
As its set in my head now, after the capture of Charles I the second time (when they executed him), General Henry Ireton decided that Charles I needed to be killed, and directed Pride's Purge. With the stronger Leveller presence in the NMA, the Parliament, instead of being purged by Pride ala OTL, is forced to enact the radicialized Heads of Proposal that was accepted ATL, and dissolve itself, with new elections to come.
The NMA "clarifys" the Parliment's approval of the radicialized Heads of Proposal and enfranchises soldiers. The Army Council also intervenes in deciding who can and cannot run for Parliament. Lilburne's Levellers are the best organized party in the country, and dominate London, which due to the redistricting now has significantly more seats. When elections come, Levellers come out as the strongest single party, and for some reason anyone with Royalist sympathies seems to have been unable to run (or doesn't win if they do).
The Lilburne Parliament (so-called because of the election of John Lilburne, the intellectual driver of the Leveller party) acts to get rid of the King, with Lilburne and his faction driving proceedings. The group of judges who decide to execute King Charles demonstrates the alliance that the NMA and civilian Leveller leadership have made, with John Lilburne and Henry Ireton both sitting in judgement of the King.
Once the King is executed, the NMA turns to reducing Ireland (an effort commanded by Henry Ireton) and then fighting the Scots (commanded by Cromwell) when they invite Charles II to lead them. In England, Lilburne rules through Parliament, pushing through key points of the Agreement of the People.
The fighting of the Third Phase of the English Civil War goes basically according to OTL. Charles II is killed in the final battle against the NMA, leaving the crown to his younger brother, the 18 year old (and staunch Catholic) James II (The First Pretender).
Since Pride's Purge didn't happen, and instead new elections were called, which were dominated by the Leveller faction, once Charles II dies, Parliament is able to rule. Though operations continue in Ireland, the NMA begins to be demobilized, with troops being settled either on seized Royalist property in England and Scotland, or (much more) in new settlements in Ireland.
Cromwell, though he is uneasy about Lilburne's faction in Parliament, oversees the last operations in Ireland, before retiring from the Army. In 1654 he is elected to Parliament. In Parliament he leads a new faction of more conservative MPs, who end Lilburne's control of Parliament and attempt to create an executive authority. In '56 Henry Ireton introduces the "Council of State Act" which would create an executive body composed of MPs elected by the Parliament. The Council would be elected at the beginning of every new sitting of Parliament. Executive power would rest with the Council of State, and the Council of State would rule when Parliament did not sit. The leader of the Council was the "Lord President of the Council of State" (shortened to "Lord President"), a position that Oliver Cromwell was elected to in '56, re-elected to in '60, and which he died in ('61).
After the radical acts of Parliament from 1649-52, during the last stages of the English Civil War, the arrival to Parliament of many former NMA officers serves to counter-balance things. The same party names that had become familiar during the Civil War the "Grandees" and the "Agiatators" come back into use, to describe the factions led by Henry Ireton and John Lilburne.
With the Long Parliament's approval of the radicialized Heads of Proposal, the biennial elections of Parliament (with the exclusion of Royalists and their sympathizers) provided a more even keel for the Republic to operate from. The radical program passed by the Lilburne Parliament (49-52) actually got wide-spread support and participation, and granted greater legitimacy and popular support to Parliament and its acts.
Good news for the Stuarts as they will get plenty of backing for their attempts to retake their kingdom. Whether they would be successful is another matter.
I don't think James II, the Most Catholic Pretender, is going to be a big hit in England . . .