Or, having a Leveller on top of the New Model Army. And at the same time, Leveller ideology keeps spreading unchecked both among the soldiers and the common folks (IOTL, by 1648, they received support from like a third of London citizens despite only emerging in around 1646) - and having a leading Leveller commander on top of the Army would have just accelerated its spread. In the end, a mutiny probably wouldn't have been necessary. And, not just the NMA, even the Trained Bands were rife with republican/radical tendencies IOTL.Without Cromwell you may get a Leveller mutiny that overthrows the grandees. After that it's mostly a matter of the new model army suppressing revolts till the end of time.
Of course this will radicalize the country. Now combine this with the Leveller spreading unchecked stuff above and you would have a more than sufficient POD.To do that you'd need a longer bloodier war, which would radicalise the moderates enough to support a republic.
A big bonus if you can have just a few Spanish/French battalions showing up on the Royalist side - calling upon foreign support in a civil war is a surefire way for the Royalists to lose power/support very fast, especially if these guys happen to be Papists. Further bonus if these foreign troops behave like every combatant in Ireland in the same period.
You can also conveniently kill off Charles II and James II with a strayed cannonball - both showed up in Edgehill and Charles more than a few other battles IOTL.
In the long run, well, the Levellers would have certainly established a Puritan school system (not unlike the one in New England) to educate children into good Republican citizens. England at that time did not have a school system, so it is a matter of building a new one rather than replacing an existing one (mind you, the latter is more difficult).
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