One model to consider is the Dutch Republic- in existence in the same continent and era, not a monarchy, stable over a long period of time. With that in mind I see no inherent reason why some form of the Commonwealth of England couldn't last, if it survived the immediate aftermath of Cromwell's death.
As a sidenote, a long-lasting Commonwealth might end up less democratic than Britain turned out in the long run- Cromwell maintained tight control over rarely summoned parliaments, and everyone inside the system seems to have just accepted that his son would succeed him- seems like a recipe for a hereditary dictatorship tightly allied with a large standing military.
As for what happens after a couple of generations of Cromwellian rule- does a Cromwell just take the title of king? A Stuart restoration? Some other monarch conquers England citing dynastic connections? England becomes a democratic republic in fact as well as name? The American Colonies become the real heart of the Commonwealth and England becomes their irrelevant appendage? Frankly basically anything is possible when you start considering the sheer volume of butterflies that this POD throws out. Charles II reign gutted the English military (in order to avoid coups) and established the hearth tax (in order to try to finance his lavish lifestyle) and the Royal Society (because science was trendy) and actresses on the English stage (idea imported from France so Chuck could sleep with them) so that's the military, financial, scientific, and cultural spheres of England specifically and Europe generally that will be seeing some fairly dramatic butterflies within a generation.
