Lots of good points raised here: not only is preserving the Commonwealth perfectly possible, but there were indeed several times during the Age of Revolution (clue's in the name...) where Britain was wobbly.
Of these, the best candidate in my opinion is the early 1830s, coinciding with the successful Belgian and French revolutions. In terms of results, indeed, 1832 was more-or-less the British Days of May by other means.
And it was a time of crisis: in politics, where the rising but young, insecure, and virile capitalist class seriously contemplated organising a run on the bank to sabotage any reactionary regime; in the cities, as Chartism began to stir; on the land, as England's labourers smashed the machines for a second time; in Ireland...
It took all the efforts of a moderate king and a subtle PM to keep it together. So I'd cross this over with Wolfpaw's suggestion and give us an anything-but-moderate king. It's as simple as shooting William IV when he's heir during the Napoleonic Wars. If the other brothers still predecease him, the thoroughly vile Earnest Augustus has a clear shot.