Well first off what Cromwell established was a republic, for the simple reason that it had popular sovereignty.
Charles the First was executed on the authority of the people of England.
And second no, not really.
Post Cromwell there certainly were times when the British monarchy was unpopular, George the Third, the late Victorian Era, Edward the Eighth and immediately after the death of Princess Dianna. But always, a new monarch was preferable to no monarch at all.
This is the great gift that Charles the Second left his successors, the freedom from blame. The reason a French Revolution style revolt taking place in Britain is downright impossible.
Because the blame could always go to parliament not the monarch, it freed monarchs from political responsibility and focus on being beloved by their subjects. The only British Republic scenarios I can think of are ASB ones like Edward VIII staying king and becoming a dictator or the Queen ordering the death of Dianna.