This is a question I've never heard asked about the English Civil War (+1 for thoughtfulness and creativity!), but perhaps for the reason that it's not relevant to the conflict. The overwhelming present-day historical consensus holds that the tensions and causes of eventual warfare between Charles I, Parliament, and the realms over which they ruled were homegrown. In short, they had as much to do with Charles' personality and horrible crisis management abilities as the political and religious ideals of Parliament and the Roundheads. The form of "Republicanism" that emerged under Cromwell--and the constitutional monarchy formed after the Glorious Revolution--were sui generis to 17th century Britain, as were the Protestant faith traditions that underpinned them...
No political incenctives for intervention
As the OP and commenters have pointed out, the military conflicts that were sapping the Continent at the time meant that there was absolutely no incentive for foreign intervention from polities that--in times of peace--would have had an interest in stirring the English pot. (I'm referring to the Dutch, France, Spain, or Sweden.) Consider as well that just because England was a "Republican Commonwealth" under Cromwell does not mean that the Dutch or semi-democratic city-states of Italy considered it worthy of ideological support or even lip-service, in the way that America today might provide to other liberal democracies.
A Protestant Civil War
It's also important to note that from a religious standpoint, this was very much an intra-Protestant conflict. The factions had no sympathetic parallels on the Continent. Charles I subscribed to "High Anglicanism", which may have been Catholicism in form and function, but hardly in affiliation. Hence no Catholic countries would have been religiously motivated to support the Royalists. Conversely, the Presbyterian Scots and Puritan Roundheads adhered to largely grassroots, English sects of Protestantism. Sure, they shared some common doctrines with German/Swedish Lutheranism, the Dutch Anabaptists, or France's embattled Huguenot minority, but not in ways that would underpin meaningful political support or a productive, cross-Channel military alliance.
A word on schools of interpretation
There have been a lot of competing schools of interpretation concerning the English Civil War's ideological motivations over the years. Describing them here is outside the scope of the question, but I mention them since most have at one time or another claimed ideological sympathy with contemporary thinkers on the Continent. A good example is Thomas Hobbes' commentary that the war was a clash between those who ascribed to an emergent scientific worldview and the forces of traditionalist religion (see the "Behemoth" excerpt from Robert Kraynak's History and Modernity in the Thought of Thomas Hobbes). Hobbes openly expressed that his view was shared by other intellectuals both in England and abroad.
However, let's take a step back and observe that the current "overwhelming historical consensus" on the conflict's causes that I allude to in the first paragraph is scarcely 20 years old. Any theories devised earlier have been more or less discarded, whether for imposing values upon the conflict's factions that are anachronistic (in the case of the Whig School), incompatible with 17th century English societal dynamics (see class-based and Marxist interpretations), or too personality-driven (i.e. Hobbes).
Unfortunately for Hobbes, and the Whigs, Marxists, and Revisionists who came after him, most professional historians today hold to an interpretation that puts the Crown's difficult relations with the polities of Scotland, Ireland, and "Parliamentary" England front and center.
