There is very little republican movement outside the Levellers, and what is is even further away from the mainstream- Ranters, Fifth Monarchy Men, a shower of millenarian loonies for the most part.
Parliament wasn't Republican in any credible sense, at least not more of it than the tiny fraction which was still there at the end when it came time to sign Charles' death warrant.
It began as a faction fight among the establishment for them, and a high proportion thought it had gone too far by 1647. Look at the attempts to come to some kind of accommodation with the King.
What is happening is that an initially religious conflict is putting down pragmatic roots; almost no-one in England or the (dis) United Kingdom as a whole began as a republican except the Levellers, and they too took their inspiration from biblical roots, with some folk memory mixed in.
Even the men who cut the king's head off, almost all (except maybe Rainsborough) began as moderate reformers with a strong religious streak. Without they are drawn outwards by that religious impulse, there's no way to get them to a republican platform.
At the historic Putney Debates, Cromwell and the rest of the leadership of the New Model present basically shouted down the extremists in favour of a moderate, reforming course, which only by accident and repeated proof of bad faith on Charles' side broke into open regicide.
Expedient followed expedient after that, and Cromwell- who had trained as a lawyer in his youth, his father's death had meant he inherited without finishing his degree- could think of no other means of holding the law of England together except that which it was already shaped around, a head of state- a king, or at most a placeholder for one. He had no plan and no platform, save blundering through.
A republican successor is somewhat pointless; the entire point of doing away with the king and listening to the extremists would be that the system changes. There would be a new parliament called, on the expanded franchise as insisted on by the Levellers, and who that would elect- who knows?
Bet on Ned Ludlow, maybe. He was one of the few Levelling- minded men who remained on good terms with Cromwell, his second in command in Ireland (which was a popular cause in England at the time, now matter how immoral the facts on the ground call it out as being), and a politician and a survivor. He would be a good bet for the second Lord Protector of the Commonwealth, and the first Protector to actually be elected.
Butterflies, though. Look at the Anglo- Dutch Wars, intervention in the fronde. England was a military republic, and would have it's wars- so much of Charles II's reign happened the way it did because he was locked into a course started by the commonwealth, not the other way around; which he tried to mitigate and bring to a neat end, but could not quite achieve.
That would only continue and intensify. You're as likely to get a Second Hundred Years' War.