The short answer is - because a revolution was terrifying to the ruling class, and they feared the same thing would happen in Britain if the French revolutionaries were not completely crushed. I'll give the longer answer below.

On your point about the English Civil War, Charles I was beheaded and replaced by a Lord Protector, but the parliamentarians ultimately lost. In 1660 Charles II was restored, and the Restoration formed the basis of a new English monarchical culture that had very much been adopted by the 1780s. The Civil War period was called the Interregnum because it was an aberration, it was not celebrated by the aristocracy who were responsible for British participation against the French Revolution in the 1780s.

On Anglo-French relations. Yes, French was Catholic and a rival, but France and Britain were closely connected in many ways. Britain relied on France for trade - for example, Britain did not make enough paper to support it's book trade, so used French imports. British culture of the eighteenth century was strongly influenced by France. Members of the aristocracy would've been fluent in French, and a substantial proportion of reading material available was either direct French import, or translated from French. British culture was also appreciated in France. 'Anglomania' of the mid-eighteenth-century saw British cultural products consumed and translated feverishly in France. British people who had disposable wealth could travel to France recreationally, and did so. Young British gentlemen would take 'The Grand Tour' as part of their education, which involved them travelling through Europe, and spending much time in France, in order to appreciate the history and culture of other European nations. In short - France and Britain were rivals, but were connected in a number of ways - cultural and intellectual shifts in one country would affect the other, and so revolutionary fervour in France would be concerning to the French-speaking, Voltaire-reading British upper classes who were intimately tied up with French culture.

There are broadly, then, a few reasons why the Britain became so involved in the counterrevolution. First, as I've mentioned, was the possibility of revolution in Britain. Radicals, including republicans, had existed in Britain for a long time, but they were emboldened by the revolution of 1789. You can see in radical writing of this period from people like William Blake, William Godwin and Mary Wollstonecraft that seeing a revolution in Britain's closest national sibling was inspiring. Many of the ideals that British radicals held: human rights, individual liberty and freedom of religion for example - were being enacted as a result of successful revolution. This emboldened British radicals, and there was an explosion in blasphemous and sedition print and speech around the revolutionary years. For the aristocracy and the monarchy, an emboldened British radical wing could threaten revolution at home. One way of ending that fervour was ensuring the French revolution was ultimately unsuccessful, by crushing it militarily.

This, though, doesn't fully explain why Britain went to war in 1792 rather than 1789. The execution of Louis XVI in 1792 and the following terror was a major shift for British onlookers. Conservatives like Edmund Burke reported with horror on the daily execution of aristocrats and arbitrary violence in the street. Many of the radicals who had celebrated the revolution of 1789 were horrified at what was described to them in 1792. The aristocracy of Britain could see people they felt affinity with, whose culture they were in some senses raised in, being executed just across the Channel. Much of the writing by British anti-revolutionaries of this period is emotional, and the argument for participation in the war against the French was made on those terms. Those who commanded the British army were fighting in solidarity for the aristocrats they felt affinity with. I think a kind of upper-class solidarity is an appropriate frame through which to view the choices made here. The French regime after 1792 was potentially destabilising for Britain, and its continuation was personally and emotionally distressing for the British ruling classes, ending the French revolution was seen as necessary for preventing a British one.

I am happy to give specific sources if requested but I would recommend Jon Mee's book on print and radicalism in Britain in the 1790s https://www.loc.gov/item/2020715215/ and Matthew Grenby's book on British conservativism and the French revolution https://www.worldcat.org/search?q=no:928431275. I'm aware this is a very culturally focused answer, but I am a cultural historian, others may find other reasons were important!