Huge question, not least because the French Revolution proceeded in several stages. Only comparatively late in the game did it become anti-monarchical, and by that stage every European power was basically opposed to the French revolution, fearing that the revolutionary contagion would spread to their territories. For example, the Russo-Polish-Austrian partition of Poland was justified by Catherine the Great on the grounds that she was fighting "Jacobinism in Poland" - a rather dubious claim, but one that shows just how negatively the European powers felt about the Revolution. Many of the European powers identified strongly with the French royal family - for example, the Austrian commander in the Rhine issued a declaration to the French people warning them that, if they harmed the King and Queen, they could expect harsh punishment at the hands of victorious Austrian armies. (This is despite the Austrian Imperial family and the French royal family generally having pretty poor relations, so it shows that the anti-republican feeling was strong enough the Austrians were prepared to put aside this longstanding rivalry in the face of a greater threat). Ironically this backfired, since the Austro-Prussian armies proved incapable of making it to Paris, and it strongly discredited the King as an ally of hostile foreign powers. The more powerful European monarchy's attempts to violently overthrow the revolutionary government faded into the beginning of the Napoleonic Wars. After Louis XVI and his family were executed, his younger brothers were hosted by the Russian, Prussian and eventually British governments with the aim of restoring them to the throne and undoing the Revolution. Monarchs who weren't strong enough to actively attack France chipped in too - for example, the King of Sweden sponsored Louis XVI's disastrous attempt to flee Paris (another own goal).

However, I should qualify this by saying during the earlier stages of the revolution, at least in Britain, there was some optimism about the French revolution, since it was believed (with good reason) that it would result in France becoming a constitutional monarchy similar to Britain.

As to whether or not there was an organised revolutionary party in England, no. There were a few isolated sympathisers but nothing compared to the Jacobins. The only place in Europe where there was any sort of indigenous Jacobin equivalent of any significance was the Netherlands. As I mentioned above, there were many monarchs who insisted that there were indeed local Jacobin conspiracies at hand, but in most cases this was either scaremongering or misidentifying disparate movements (like the Polish revolutionaries, who were really constitutionalists.

Source: Buchez, Phillipe-Joseph-Benjamin. Histoire parlementaire de la Revolution Francaise, ou Journal des Assembles Nationals depuis 1789