Well the Commonwealth as its known had three real non-english enemies. Ireland, Scotland and The United Provinces.

Firstly Ireland. Prior the the civil war Ireland had been of course a subject of England and Scotland with Charles being separately considered the King of England, Scotland and Ireland. Now the Irish rebellion actually began before the civil war and it was actually on who's authority the army being raised to go fight the Irish rebels was under and the fear that Charles was going to use the army he was raising against parliamentarians that triggered the civil war. While in the later stages of the three conflicts that make up the civil war Ireland ended up declaring for the king but clearly this was out of convenience in looking for allies to bolster their cause rather than any particular love of monarchy.

. In the first stage of the civil war the Scottish Covenanters were more or less on the same side as the English Parliamentarians and in fact began rebellion against Charles' rule during the Bishop's wars before the English did. There were Scottish Royalists but they were defeated fairly soundly and at the end of the first section of the English civil war the Scottish and handed Charles over to the English after he surrendered himself to them at the end of the conflict. But during the second stage of the civil war Charles attempted to play the Scots and English off against each other in negotiations (at this point neither side intended to kill Charles for the most part) and in the end the Scottish Covenanters got offered the religious reforms they were after by Charles while Parliament was reluctant to implement Scottish Presbyterianism in England. And so the Scots fought the English Parliamentarians. It could be said that the Covenanters were more distinctly fighting for monarchism in the third section of the conflict as they fought for Charles II after Charles I was executed at the end of the 2nd civil war.

The third enemy of the Commonwealth was the United Provinces, also known as the Dutch Republic, as you can guess this had nothing to do with monarchism, rather it was a conflict over navigation rights to English waters in the Channel, trade and fishing rights. It was a purely naval war and had a fairly limited scope without any real ideological dimensions.

Concerning the continental monarchies there's a number of reasons why they didn't really care all that much.

Firstly the Commonwealth wasn't as aggressive as the early First Republic. Early on the French had been looking to expand to the edge of their natural boarders as they saw them with the aim of taking places like the Austrian Netherlands (more or less what is today Belgium) and there were calls amongst many to wage war against the Monarchs (specifically the monarchs) in order to export the revolution so to speak. The commonwealth on the other the odd dispute over marine rights but nothing so ambitious or ideologically driven.

In the reverse the war of the first coalition was at least partly driven by the coalition member's own territorial ambitions. The Prussians who were the first to declare war wanted to take certain territorial assets under french control in the holy roman empire and further expand its influence and prestige in the German states. Britain was at least considering annexing Dunkirk to have as a base. In the throws of revolution it was assumed France might be a relatively easy target. Then of course Francis II who had just come to the throne (and perhaps felt he had something to prove) was the nephew of Marie Antoinette so he had something of a personal connection to the French monarchy. Austria having already been brought to the edge of war over aggressive, brinkmanship style negotiation on both sides of the conflict during his father's reign.

Additionally England wasn't as important in the mid 17th century as France was in the late 18th century. England's colonial empire was fairly small, certainly much smaller than the Spanish. Likewise its army was behind the times prior to the civil war and its navy was not exceptional. It was relatively urbanised but not to the extent it would grow to be. England wasn't really one of the true great powers between the end of the 100 years war and the 7 years war. France on the other hand while of late it had been in serious economic problems and had lost the important 7 years war but it was still considered a considerable power and as a continental power was inevitably much more entangled with continental politics than England was in the mid 17th century.