From the perspective of both constitutional law, and practicality, they have not yet lost power. However, I am unsure if this sub-reddits rules allow for an overview of the current powers of the monarchy.
The short answer would be with the parliamentarian victory of the English civil war, and the glorious revolution with the installation of William of Orange.
However, as a legal historian, the unique character of the English (and later, British) constitution must be recognised. Unlike most countries there is no document detailing the working and power of the various branches of government. Instead, shifts in power are governed by convention.
The crown has various powers: these stem from three sources; statute, prerogative, and 'the third source'. Perhaps the most potentially significant is the ability to refuse assent for a bill, without the royal assent parliament cannot make law. However, the practical power of this is tempered by the strong convention that it will not be exercised. Indeed, the last time a monarch refused assent was in 1707 when Queen Anne vetoed the Scottish Militia Bill. The last time the Monarch dismissed a government was in 1843. It is an open question whether these prerogative remain law after centuries of disuse, or whether the convention that they not be used has chrystalised into positive law. Practically speaking, it would be controversial for the monarch to exercise some of her more extreme powers, and parliament probably wouldn't stand for it. However, it is worth noting that the queen has exercised royal power in modern times through her representatives in the forms of the Governor Generals of NZ, Canada, and Australia.
It should also be noted that parliament has usurped the power of supply; that is, the crown cannot raise or spend public funds without the consent of the house, and the judiciary has in 1610 asserted the power to judge whether royal actions were lawful (case of proclamations), and recently found that royal prerogative could be judicially reviewed. (CCSU case).
Basically, there's been a Millenia long, slow abrogation of royal powers starting with the Magna Carta, sped by the English Civil War and Glorious Revolution, and the evolution of the monarchy (in part, I believe to avoid the fate of most absolute monarchs in Europe post enlightenment). This dwindling of power is still ongoing.
Sources: Texts- Dicey's Law of the Convention, O'Hood, Phillip and Jackson Constitutional Law.
In more detail, the power of the British monarchy had risen and fallen many times over the centuries, with “the early stirrings of a concept of consent to rule” (A Short History of England), beginning all the way back in the 890s, when King Alfred ‘the Great’ wrote legislation that bound kings to ensure that law and security were upheld. The Magna Carta written by King John in 1215 is often hailed as the watershed moment in limiting the English monarchy’s power, but, other than getting King John out of a sticky situation with his land barons, the effect it had at the time is debatable.
As far as I’m aware, the next developments of note happened under Charles 1 reign in the 17th century. The first half of the century saw much hardship in the British Isles, largely due to weather related famines, but exacerbated by Charles’ propensity for foreign wars. A few other poor decisions, notably his attempt to change England and Scotland’s liturgy, led to the English civil wars in the 40s, culminating in Charles’ regicide in 1649. However the successes of the English republicans weren’t to be long lived, and Charles’ son, Charles II, was invited back to rule in 1661. He was able to revoke a lot of legislation passed by the republic, and gained most of the powers his father had lost (in the time leading up to his execution). And he ruled successfully without Parliament for the last five years of his reign.
However upon his death in 1685, his Catholic brother and successor James II made the same mistake as Charles I when he tried to tamper with the kingdom’s liturgy. This again annoyed much of the populace, and a small group of peers invited Prince William of Orange (and James’ son-in-law) to invade England. Remarkably, just four months later, in November 1688, William had pulled together an army and did just that. Although not welcomed by everyone, he faced little resistance, and, largely peacefully, removed James from the throne. Following lengthy and difficult negotiations, William and his wife Mary became joint monarchs of England, and in return guaranteed certain liberties. They remained in charge of the kingdom’s military, but only for six months at a time, meaning they had to recall parliament frequently to renew it. And most importantly, Parliament gained control over the countries taxation and finances.
Sources: A Short History of England by Simon Jenkins, and Global Crisis: War, Climate Change & Catastrophe in the Seventeenth Century by Geoffrey Parker
To add a question on to this - were there stages to the loss of actual power (i.e. they lost political influence, which started the decline of military power etc.), and how (if ever) were these 'gaps' filled?
hi! you'll find more on this in the FAQ