In short, because the means of power were dictated by democratic means.
During the time of Oliver Cromwell's rule England was a commonwealth that can be considered a republic because the people were represented in the government by elected officials. After the defeat of Charles I in 1653 the victors drafted the Instrument of Government (full text here), and afterwards Oliver Crowell was declared Lord Protector of the realm. However, the legality of the Instrument of Government was doubted, and in 1657 another group drafted the Humble Petition and Advice (full text here), which declared Cromwell Lord Protector for life, and gave him the ability to name his successor which is how his son, Richard Cromwell, became Lord Protector upon his father's death.
In sum, the "people" of England democratically set up the means by which Oliver Cromwell, and later his son Richard Cromwell, wielded their power. So, although the Cromwells exerted tremendous power they were propped up by England's two written constitutions.