Agreed. Radicalism of that sort is damned un-British.
Normally, it goes like this for the Brits:
"What do we want?"
"Gradual change!"
"When do we want it?"
"In due time!"
I know you're probably joking to some extent, but its important to remember that radicalism has a huge and important place in British society at the time.
The naval mutinies at Spithead and the Nore need to be seen in a context of the 1790s. The French Revolution, far from being seen as something horrifyingly alien, continued to galvanise some sections of British society. You need to read EP Thompson's classic
The Making of the English Working Class on this. Thompson uncovers radicals from London Corresponding Societies [so called because they were communities of like minded people who communicated by letter with other such groups around the country] in amongst the mutineers.
page 185: In June 1797, shortly after the mutiny, a certain Henry Fellowes was apprehended in Maidstone distributing handbills to the troops. He was an emissary of the London society and...reported two divisions of the Maidstone society active [with sixty in attendance] and ordered more handbills particularly for the Irish soldiers as well as copies of Bonaparte's Address and Paine's Agrarian Justice.
Remember Ireland is in revolt the following year, 1798, agricultural wages and prices are in flux because of the war, and 1790s society in England is not completely different from the conditions that helped spawn popular support for the French Revolution across the Channel. Especially with the Prime Minister cracking down on those civil liberties often thought to be the distinction between British constitutional monarchy and the Ancien Regime - such as suspending Habeus Corpus in this period.
I'm not claiming that there was a seething majority ready to overthrow the status quo in 1797, but it wouldn't be ASB to see it happen by any means.
Everything is un-insert-nationality-here until it happens.Russia going communist, France being torn apart by the Revolution etc.
The issue is how isolated the incident is. If it had influence outside its area, if it had a somewhat coherent leadership high up and if the society allowed it then it would be of consequence.
Essentially that.
Just because Britain hasn't had a revolution in its modern history doesn't mean that it is inherently un-radical. Just that those passions never quite boiled over into revolution which, to be honest, is only one form radicalism can take. Take a look at the fever-pitch of the 1830s and 1840s in Britain if you don't believe me and tell me the social tensions and pressures aren't there. Britain could easily have tipped over the edge into revolt in that period.
As for what this changes, yes, well, everything. Modern British identity is forged in the 1790s and 1800s in opposition to France [this is where you get the myth of a stable and non-radical Britain first emerging]. But also the isolation of war, cutting Britain off from the continent for large periods of the next few decades, was actually good for British development in economics to some extent. If Britain goes through a decade of social turmoil because of a revolution spreading from these mutinies, you'd see the birth of the railways knocked back, industry developing more slowly, and the pace of the industrial revolution grinding down as investors flee or refuse to take risks. You might see Britain eclipsed in India, you might see the US willing to roll the dice about Canada, more slave revolts in the Caribbean, etc etc etc.
The period was a defining moment for Britain, and you're essentially throwing the pieces up in the air. Its hard to predict how they would land, but they certainly won't be the same as OTL.