and figure out why it survives
England would likely become the same world power - based on trade and industry - as in OTL. The same rights (a Bill of Rights of the Commonwealth) or maybe more progressive would be enacted by the republican parliament. The colonies in North America would become more republican as they already was.
Conflicts would erupt in regard to the new republican constitution - a monarchical republic under a more capable Lord Protector than Richard or a more aristocratic republic centred around a council/parliament.
Also, democratic agitation is conceivable and inevitable after foreign examples of popular uprising. A lasting Commonwealth of England would also have to deal with riots and rebellions in Scottland and Ireland; also, absolutist powers on the continent would have to be confined by ideological or military combat.
Simultaneously, England could support revolutionary movements in Europe (like a second Fronde in France or liberal groops in Sweden) to promote their order and install puppets/friendly regimes while fighting against the main concurrent on the See, the United Provinces, even if they are actually a republic).
Finally, the American provinces would mabe remain in the posession of England: a republican parliament, beeing more candic towards a representation of the oversea territories.
The Levelers were a political movement during the English Civil War that emphasised popular sovereignty, extended suffrage, equality before the law, and religious tolerance, all of which were expressed in the manifesto "Agreement of the People". They came to prominence at the end of the First English Civil War and were most influential before the start of the Second Civil War. Leveler views and support were found in the populace of the City of London and in some regiments in the New Model Army.
Maybe a second democratic revolution in a few decades, based on the ideas of the Levelers?
So as often happens, the English Revolution would proceed in stages, first a republican revolution, followed by a democratic revolution.
In OTL, they were brutally suppressed by the Commonwealth. Perhaps Cromwell's successor is more sympathetic to their ideals and allows then to resume operating?
us style constitution by 1750.
"US" in OTL or in ATL? Do you think there would be Americans wanting the independance of the thirteen colonies?
Well, Cromwell was basically the Puritan's wet dream (in fact, a lot of New Englanders returned to England to fight as Roundheads in the Civil War), but there's also the fiercely Cavalier Tidewater and Carolina colonies. The Quakers and Germans in the Delaware didn't much care either way - but the Puritans in general didn't much like them. It's possible that a continuing Commonwealth might marginalize the Southern colonies and antagonize Pennsylvania and West Jersey enough to get them to want to withdraw.
Coolest possibility: English monarchy in exile in Virginia.
"US" in OTL or in ATL? Do you think there would be Americans wanting the independance of the thirteen colonies?
otl US, the enlightenment would still occur, and the classical liberal principles that gave birth to the US would still emerge