I know only a little bit about the English Civil War and Oliver Cromwell. Has there been any attempt at republican government or significant republican sentiment since those events?
OK, good morning! This is a very interesting question and I happen to study British radicalism in the 1790s. So I'll do my best to give you a quick overview.
Radical organisations had existed in Britain prior to the French Revolution, notably the Westminster Association of 1780 and the Revolution Society which was formed in 1788 and claimed to champion the principles of the Glorious Revolution of 1688. These societies believed in constitutional reform rather than outright Revolution. Many of the publications of the radical societies at this time demonstrate that their demands were hardly what we would nowadays describe as radical.
A 1792 pamphlet issued by the London Corresponding Society (the most influential radical organisation of this decade) contained demands for annual elections and universal suffrage and expressed the belief that "the House of Commons ought to be restored to that state of Independence in which is was settled at the Revolution". The pamphlet also resented that anyone holding these beliefs was "stigmatised as a Leveller and an enemy to his King and Country"^1 ^2 . In general, the evidence shows that in the first half of the 1790s most British radicalism was not republican. The desire was to reform the existing constitution rather than overthrow it.
The key ideas of British radicalism at this time can be found expressed in Thomas Paine's The Right of Man, which is the seminary text of late 18th century radicalism. In Making of the English Working Class, E. P. Thompson claims that Paine was more important for British radicalism than the French Revolution and the "Paine established a new framework within which Radicalism was confined for nearly 100 years"^3 . Other authors, such as H. T. Dickinson, point at that although Paine advocated a democratic republic, very few of the radical societies went that far^4 .
The turning point for British radicalism came in 1795 when the Pitt government passed the Two Acts. The Seditious Meetings Act prevented the radical societies from holding public meetings and helped the government to destroy the effective leadership of the societies by prosecuting, harassing and intimidating the leaders^ 5. This meant that, by mid-1797 the more moderate reformists had either been arrested or driven out of radical politics and been replaced by a "violent revolutionary fragment"^6 . These hardened revolutionaries were far more prepared to resort to violence to install a republic, though they never achieved the same levels of popular support that their predecessors in the London Corresponding Society had enjoyed prior to 1795.
The radical societies were eventually outlawed in 1798 and 1799 and this pretty much ended the British reform movement until the Benthamite movement of the mid 19th century.
So in summary, though there was a significant radical movement in the first half of the 1790s that movement was reformist rather than republican. It was only once the societies began to be repressed that they became more republican in their outlook and objectives but by then they could not amass enough popular support to pose a significant threat to the established order.
^1 H. T. Dickinson, British Radicalism and the French Revolution 1789-1815 (Oxford, 1985) p. 13.
^2 National Archives: TS 24/3/45
^3 E. P. Thompson, The Making of the English Working Class* (London, 1963) p. 94.
^4 Dickinson, British Radicalism and the French Revolution p. 14.
^5 ibid pp. 41-2.