I'd say three times, and some kinda-sorta examples:

The first time would be in the run-up 1832, prior to the passage of the Great Reform Act, which saw a vast increase in working and middle class political participation, especially in the cities that had sprung up as the Industrial Revolution began to get underway. Not only did you have vast legal political societies, such as the Birmingham Political Union, but there were also violent riots when the Second Reform Bill was rejected by the Lords in 1831, and open defiance of the law when the National Political Union was declared illegal and kept going anyway. When the Third Reform Bill is rejected, it leads to the famous Days of May, a period of national political agitation, rioting, mass demonstrations, bank runs, and the spectre of armed insurrection. Eventually, after some rather complicated political double dealing, the Reform Act is passed. Arguably, given the mass participation, the Days of May were the closest thing Britain had to an massive revolution during the 18th century.

The next nearest that Britain comes would be the era of "Red Clydeside" in Glasgow, which marks the only time tanks have been deployed against civilians in mainland Britain. What happened in Glasgow is a period of noted political radicalism and rent strikes starting from the Singer factory strike of 1911 and culminating in a mass strike and riot in George Square in 1919. The British government certainly consider this a potential revolutionary moment - Glaswegian soldiers in Maryhill barracks are locked inside to stop them joining the risers, and Winston Churchill, Home Secretary, sends 10,000 troops from the rest of Scotland, including tanks, into the city to quell the rioting. Eventually, the workers are placated by the promise of a 47 hour week.

The third time would be the Chartist movement, a working-class mass movement that lasts from 1838 until 1858. Now, the Chartists are explicitly reformist, not revolutionary - they want manhood suffrage - but occasionally they do spill into violence, such as the Newport Rising. On other occasions, such as the rejection of their 1842 petition, signed by 3m+ people, the mood became ugly enough that a revolution could have been possible.

The other "kinda-sorta" times: one would be the Jacobite rebellions in Scotland and the North, but these are sectarian and nationalistic rather than revolutionary - its a dynastic dispute over who has a right to rule.

Another might be the Irish uprisings, but again: nationalist rather than British, and prior to the Great Famine not universally supported.

A third might be the English Civil War, which was certainly revolutionary in the sense that it involved the killing of a king and the proclamation of a pseudo-form of republican government - although in practice it was a military dictatorship. But that's a very complicated period in history and not one I really know much about.

Personally I'd say there never really was a time where revolution was looming in Britain. However the time where a large scale uprising was most likely; was in Glasgow and the surrounding regions on the Clyde in 1919. For years you had an increase of radicalisation in the urban parts of the central belt, with militancy further fueled by opposition to the war, rising rent during the war and much more.

On January 31st 1919 you had what was known as the 'Battle of George Square', a large strike in Glasgow commonly known as the '40 hours strike' turned violent, with the police force against 60,000+ protesters involving key figures on the Scottish radical left, and trade union leaders. Things were only calmed with military intervention by the government, soldiers were deployed from elsewhere in Scotland due to fear of local men joining the protesters. 10,000 or so men were deployed to the city including machine gun posts, a howitzer and medium tanks to try and end the violence.

Although the Kingdom of Great Britain, or later the United Kingdom, did not yet exist the English, Scottish, and Irish held a parliamentary revolution in the 1640's against the monarchy, known as the "War of the Three Kingdoms" or the English, Scottish and Irish civil wars.

Charles I had effectively been enforcing taxes and religious laws in each country by undermining each of the parliaments, and they were all tired of his shenanigans. Scotland went to war first, with citizens rioting, then Ireland, then England.

The result was the monarchy calling their own forces and a civil war broke out across the islands for nearly a decade. The monarchy was deposed, the king was killed, and a republican government was established. The republican government did not really work with how the English (and Irish) were governed, and the monarchy was restored.

40 years later there was another revolution in England, Scotland, and Ireland. The parliament of England were to depose the king, and invited his closest relative, William of Orange to succeed him. He brought a fairly large Dutch fleet to the islands, landed, and with support from the English, the Dutch fought a few battles against the kings and the king fled, abdicating the throne.

After the Peterloo Massacre in 1819,a group of 80,000 workers organised by the "Manchester Patriotic Union" held a demonstration demanding major political reforms,they where unarmed but the Army interfered,injured a few hundred and killed a smal number of them,this caused the conservative political parties to alter their strategy so as to prevent a revolt,took some reforms and supported movements like the greek war of indipendence and the revolutions in South America

The Peasant's Revolt of 1381. It lasted for six months, the rebels entered the Tower Of London and killed numerous officials, including the Lord Chancellor and Lord High Treasurer. Richard II agreed to their demands, including the abolition of serfdom, but later went back on his word.

The decades leading up to it were a period of simmering rebellion that occasionally boiled over into strikes, protests and riots, and finally in 1381 open warfare. Had the rebels not been loyal to the king, they might have killed Richard (then only 14 and with few troops protecting him) as they did many of his ministers and advisers.

Although the rebels were defeated, and none of their demands were met, both the king and the lords were remarkably circumspect in their treatment of the peasants. The expected poll tax was not enacted, and unable to raise more money, England reduced its military expeditions in France. For many years afterwards, rural workers were known to elude to the revolt when negotiating over wages and rents.