So the first thing to know is that there were very few slaves in England at the time of the Black Death - the Normans had outlawed the slavery practised by the Anglo-Saxons and slavery sharply declined from around 10% of the population (recorded in the Domesday Book) to almost nothing. You can read about that in this recent answer by /usteelcan909. There were still some slaves - the Knights Templar, for example, enslaved well educated captives and used them to cope with the vast logistical challenges of running an organisation that spanned from Ireland to Jerusalem - but we're talking maybe a few hundred at most, and they fell outside the authority of the Norman government.

In the early 14th century, most of Europe had a major problem on its hands: population growth outpacing the rate of land clearance. Surviving records indicate that from around 1100 to 1340, England had lost about half their woodland to facilitate better agriculture, but the population of England had tripled from about 2 million to 6 million. That's obviously a problem - there wasn't enough farmland for all the peasants, and they need farmland to, you know, live and stuff.

The result was what's commonly referred to as 'tenured farming', and others would call it serfdom. Peasants who wanted their own land increasingly surrendered some of their freedoms in exchange for land from noblemen. Because the peasants did this of their own free will (legally speaking, obviously there was a lot of economic pressure to do this) and because the peasants got something in return, it wasn't considered slavery. The specific arrangements of these deals varied from person to person, but generally speaking the peasant would be expected to work the land for several months of the year, and maybe do some work for the lord on the side, all for low pay. This happened across most of Europe, with the exceptions of Flanders and Holland where they had the brilliant idea of making more land with dykes so they did not face the same land shortage.

So we get to the late 1340s, and oh no half the population is dead. Suddenly, the land shortage is completely gone, replaced with an agricultural buffet. The average amount of land owned by a peasant essentially doubled in the space of two years, and many found themselves even better off as they inherited the lands of others. But that's too much work for one peasant family, so suddenly the land shortage was replaced with a labour shortage. A contemporary poet named John Gowar wrote of the post-disease peasantry as 'sluggish, they are scarce, and they are grasping. For the very little they do they demand the highest pay' - this was what 'more economic power than before' actually looked like; people felt that if they weren't going to be paid well, they wouldn't bother to work well.

In England, the nobility responded to this by passing laws which required tenured farmers to remain at their posts, as it were. They didn't plan to bring in slaves or immigrants, they simply decided to try and keep the economy in its pre-plague state by law, and force people into agricultural work. They saw the problem coming too, and were sorting out legislation whilst the disease was still ravaging the country. In 1349 the Ordinance of Labourers legally compelled people to stay at the farms. It wasn't working, and in 1351 the Statue of Labourers attempted to cap wages at pre-Black Death levels. This was drastic - it made it a criminal offence for which you could be imprisoned for 40 days to ask for or provide a raise, or to increase the sale price of goods. Later legislation introduced punishments such as branding for running away, and prison terms for refusing to work. To the nobility this was preserving the natural order, to the peasants it increasingly looked like slavery. It's fair to say that the response of the English nobility to the labour shortage was essentially to try and freeze the economy.

It sort of worked. It was badly enforced, but surviving tax records suggest that the increase in wages slowed. They were still rising though, and over the following decades the middle class swelled. Laws attempted to prevent peasants owning certain goods associated with nobility, like fine clothes and boar meat, but these were widely ignored. People increasingly ignored obligations to work the land and ran off to the cities. Some cities, most notably London, openly provided sanctuary to these runaways and forbid crown officials from entering the city to find them (which they were permitted to do under their charter, but the king's advisors saw this as an abuse of power by city council).

This put the peasantry, though by this point 'peasant' just meant anyone who was supposed to be a farmer and a large portion of them had branched out into other professions or abandoned agriculture entirely, and the nobility on a collision course. This conflict with the interests of the nobility also had the effect of increasing political awareness and participation among the peasantry, and led to the formation of a social movement to reduce social inequality and improve access to politics. In time they would become known as 'levellers'. The politics of England increasingly became about levellers vs. the traditionalist nobility. It's worth noting that there were many noblemen who did not mind the increasing wealth of the peasantry, didn't bother enforcing labour laws, and had no problem with more political participation, but they were definitely the minority.

This is where we get to your question of "Was the Black Death really as revolutionary as I was taught?". The answer is no, it was probably more revolutionary than you were taught, as in it nearly caused a revolution known simply as the Peasant's Revolt.

So in 1381, when taxes were levied to help fund the Hundred Years War, there was a plot to kill tax collectors in the South East of England. It began with the murder of John Bampton in the town of Brentwood. The sources describe the violence as spontaneous, but the level of coordination among the rebels suggests a level of pre-planning. There were two main groups, one from Kent which was led by Wat Tyler, a leveller, and another less organised group from other parts of the south east. They were also influenced by a preacher named John Ball. They seem to have been well organised and marched on London with a list of demands for King Richard II, who was only 14 and essentially controlled by his advisors. They called for a reduction in taxes, an end to labour restrictions, the total abolition of serfdom, and the dismissal of certain royal advisors. Another contingent with a very similar list of demands approached from central England. It's interesting that they had a clear understanding of who they held responsible, which demonstrated widespread knowledge of how the inner government worked. London was caught in the middle because that's where the king happened to be at the time.

To cut a long story short, Richard agreed to all demands but the revolt was crushed after Wat Tyler tried to kill the king. Parliament would later declare that Richard's agreement was invalid, but their key demands were still ultimately met. The rebels had burned the properties of hated advisors (even blowing up the Savoy Palace, then owned by John of Gaunt, the king's regent and all-round asshole) and trashed the Tower of London, lynching the chancellor in the process whilst letting other advisors go. This focussed the minds of the nobility to the scale of the problem. Although the rebellion had been defeated, and smaller revolts of a similar nature across the country had been suppressed, everyone could see that their attempts to preserve the pre-Black Death labour market was destroying the country, and leading people to ask awkward questions like 'so why do we need noblemen anyway?' Noblemen also began to realise that their misgivings about certain royal advisors were shared by the people, which emboldened them to take action.

To take stock of the situation, Parliament was called. The goal had been to sort out money for the Hundred Years War, but the agenda shifted to the fallout of the Peasant's Revolt and the dismissal of some royal advisors, though not officially for reasons related to the revolt. Although Parliament did not formally end serfdom or repeal the labour laws, they were quietly permitted to go unenforced. Officially, Parliament rejected all rebel demands, but in practice it accepted almost all of them. To save everyone a lot of trouble, many lords offered the chance for peasants to simply buy their land outright and therefore buy their way out of serfdom. Levellers also became a dominant political force, with John Northampton winning a landslide victory in the 1382 London mayoral election on a platform of reducing inequality and increasing access to the city council. Parliament also decided not to levy taxes for a while, and instead reduced their campaigns against the French. After military failures and corruption scandals over the following years, the Parliament of 1388, known as the Merciless Parliament, impeached, exiled, or killed those on Wat Tyler's list (except John, he was already in exile). So although the revolt was crushed, almost all of its demands had been met in practise if not in law within a decade. Peasants became wealthier, wage growth accelerated, serfdom steadily declined.

So yes, the Black Death was indeed rather revolutionary. The government hoped to solve the labour shortage by legally compelling people to work for cheap, this caused a revolt at the hands of Wat Tyler. With the Peasant's Revolt of 1381, the nobility realised they couldn't win and, although they weren't ready to repeal labour laws, quietly acquiesced to rebel demands. But yes, the Black Death was super important in the history of England.