This would have serious repercussions upon the near-modern period to the extent that our modern geopolitical situation would be completely unrecognizable. I think the main differences, however, would come down to England's direct neighbours and her overseas colonies. Even though Cromwell himself only governed as monarch for a single year before his death, he (and his son) directly put in place the institutions that came to define English culture for generations.
Oliver had already created a cult-of-personality around his leadership by the time he anointed himself as King in 1657. As a result, when he passed over to his third son Richard IV (1658 - 1712) government rallied to the new regime rather quickly. The pro-Cromwell loyalties of the newly-renamed Grand Model Army also aided in the turnover, and as a result the opening years of the reinstated monarchy were carved in the final defeats of Confederate Ireland and pro-Stuart forces in Scotland. Had Oliver not declared himself monarch it is likely that the fragile balance of power between those supporting the re-installation of autocracy and those the authority of parliament would have been broken and the Commonwealth collapse.
Due to the length of Oliver's reign, the second Cromwellian King - Richard - is arguably more interesting, and certainly longer! As stated, it was chiefly due to the military complex of Oliver that his son was able to seize the throne as effectively as he did; had the Commonwealth collapsed, it is debatable as to whether the Stuarts would be able to seize the throne. Certainly, Richard was concerned enough about a restoration that the famous execution of George Monck at St. James' Palace brought about a Stuart-scare. It is perhaps more likely that an elective monarchy would be introduced, with a strengthened Parliament nominating an acceptable sovereign - most likely from the Dutch.
As for the colonies abroad, the distant nature of the monarchy would remain relevant with or without the Cromwellian dynasty. Columbia will form in some manifestation, although the centrist policies of the Roundhead Party (that dominated Parliament for most of the period) would likely grate with those supporting a Stuart restoration, for example. It is likely that Commonwealth-like organization would develop in North America, and especially so after the seizure of Dutch and French possessions. It was only after Elizabeth II seized power following her father's death in 1712 that the relationship between England and North America was questioned, and further still after the Succession Crisis in 1731 (that began to sow the seeds for the granting of Colombian independence).
To sum up, I think Ireland would be in much better shape and would be the ultimate victor of this timeline. There might even be considerable Irish emigration to the New World (providing that the famines still occur as OTL). France would probably be stronger - and especially so if they aided in a Stuart restoration. Nevertheless, whilst Britain maintained naval superiority under the Cromwellian dynasty under an alternate House this might not be possible. It is even possible that another Civil War will erupt as the royal authority once again pushes too hard upon the aristocratic landowners and Parliament.