I cannot answer your question directly, but the Great Reform Act of 1832, which modified Britain's electoral system, was in part propagated by a "Society of the Friends of the People", a liberal political organisation set up in 1792 (the year of the deposition of Louis XVI) and is named similarly to the radical revolutionary republican political club in France "The Society for the Friends of the Constitution", colloquially known as the Jacobins. The Jacobins are notable for their role in the "Reign of Terror" and their pushing for the execution of the King Louis XVI after his being found guilty of treason by the National Convention.

This Society, however, was disbanded in 1794 due to anti-association legislation. Much more I cannot tell you, since it lies outside my knowledge.

As per my grade 12 textbooks, Britain during the French revolution had a vestigial monarchy, ie like modern times, the monarch didn't have a major role in decisions made unlike the french. The people of Britain gradually shifted power from the monarchy to a parliamentary system by the 1750s. The British system had 3 separate parts the judiciary the legislature and the executive. The French revolutionaries were inspired by this system and hence went rebelled. So as a matter of fact the people of Britain had no monarch ruling them that they would feel the need to overthrow them. This was not the case in the colonies however. Edit: typing on phone missing some points