Note this is an amended version of an answer asking about the Dutch, Spanish, and French, response to the Civil Wars here: https://www.reddit.com/r/AskHistorians/comments/ub9ivg/comment/iexhih7/?utm_source=share&utm_medium=web2x&context=3
A simple answer is that France had their own, more immediate problems at the time and were not particularly interested in involving themselves in what seemed a largely internal matter, when in 1638 the British Civil Wars (1638-1653) began with the Bishops' Wars (1638-1640) between Scotland and England, even as these moved through into the Irish Rebellion (1641) and then the outbreak of Civil War in England (1642), there was much happening on the European stage to keep France's foreign policy under Cardinal Richelieu, and later Mazarin during the regency busy:
The Franco-Spanish War (1635–1659), the Eighty Years War (1566–1648), and the Thirty Years War (1618 to 1648), were conflicts of much more immediate concern to them. The regency of Louis XIV for much of the start of the conflict occupied France as well.
Yes, France had "problems" with England for many years previously, with fighting as recently as 1628, but by the time of the wars, the Queen of England was Henrietta Maria, youngest sister of King Louis XIII, later aunt of Louis XIV.
Religion cannot be separated from politics in this period either and while the Church of England was nominally Protestant, Calvinist Protestantism in rebellious Scotland, and Roman Catholicism in rebellious Ireland placed a schismatic dynamic into the fray that could well have meant that France, already engaging with their own religious minorities internally (the Protestant Huguenots) and enemies abroad, both co-religionists and not, just wanted no part of.
By the end of the wars in Europe, the situation was quite different - the triumph of Parliament and the consolidation of power over the British Isles in their hands in the form of the Commonwealth meant that here was a unified and powerful militarised society. During Cromwell's lifetime it was telling that the exiled Charles II, son of the executed Charles I, had little overt European support to regain his throne, and indeed even France treated with Cromwell's Protectorate in an alliance against Spain.
Further Reading
Parker, G., The Thirty Years War (Routledge, 1997)
Damson, J., The English Civil War: Conflict and Contexts, 1640-49 (Palgrace, 2008)

An old article, but interesting is Haines, H. 'France and Cromwell' Transactions of the Royal Historical Society, Vol. 5 (1891), pp. 147-156