What if the Cromwellian republic was never overthrown?

So, England was technically a republic for a short time when Oliver Cromwell became Lord Protector of England. It was dissolved after his son, Richard Cromwell was deposed and the monarchy was restored.

What if the Cromwells retained power, or England at least remained a republic?

How would history play out differently? Would England still establish a global overseas empire? Would that mean the American colonists would be rebelling against a republic and develop a monarchist movement?