By 1840, coal had been mined for far longer than a century. In fact, Londoners had been complaining about air pollution from burning coal for far longer than one century--six of them, in fact. Especially in England, coal mining was very much already an industry and a profession in the later Middle Ages.
Evidence for the use of coal dates back even further, with specific and limited uses in mind. It appears to have fueled cremation fires in modern-day Wales as early as the Bronze Age, and Roman England saw a decent amount of use: providing heat for northern forts, possibly Greco-Roman temples before Christianization, and for use in homes and blacksmith forges in East Anglia.
But even if the term "Age of Coal" has been used to signify basically every other historical era besides the Middle Ages, it was the 13th century when England rediscovered its benefits and made it a central part of making fire. It's not a mystery why, either: the 13th century was when English people had a reason to start looking around for new fuel sources.
Rapid population growth prior to the early 1300s (yes, before the Black Death) was intimately bound up with increased use of land for agriculture as well as increased need for construction and fuel materials. The predictable result was mass deforestation, in terms of trees as well as substantial brush, and competition among the remaining lumber to serve as fuel versus houses and ships. (The papacy tried to ban the sale of timber to Muslims, theoretically fearing its military use. In practice, the "fine" they levied on merchants who violated the canon served as a sales tax.) This was particularly true in London, which was a massive city by medieval European standards.
So, why not coal? Rudimentary coal mining needed no canaries or helmets with headlights. The most basic method was locating coal seams just beneath the surface of the earth, where miners could just clear off a few feet of dirt and shovel. The geographic extent of this type of mine was rather limited, of course. John Hatch further suggests that much of the archaeological evidence here points to very local use, maybe even peasants "scratching" up a bit of coal for their homes, village smith's forge, or village ovens.
So there were various forms of pit mining. One strategy was simply to dig a shaft and use a well-type mechanism to lift coal out by buckets--and then, when the shaft inevitably caved it, sink another shaft. But just like with the pillars of bridges, and sewer cesspits, some mines reinforced the walls of the shaft/pit with rocks/bricks.
Mining wasn't the best alternative career to farming--that was milling. But it nevertheless attracted a decent number of the rural teenagers and young adults who chose to leave their village in search of...well, the typical explanation is that women would work to earn a larger dowry, and men, to earn enough money to be their share of establishing a household. (Women were not miners; however, if evidence of women as petty laborers surrounding the actual construction part of medieval construction sites can be a sign, they might have worked around mines as well.)
As for the eventual destination of mined coal, the fuel is especially associated in contemporary literature with forges and other artisanal use. Of course, industrial dirtiness was a common trope (especially with dyers and butchers), and London in the late Middle Ages was intimately familiar with the dirtiness of coal burning. In the 1280s, the city set up a committee to "investigate" the problem. In 1306, they finally turned out a measure to limit the use of coal (which was essentially "ban it"), but this worked as well as you might expect (which was essentially "it didn't.")
This isn't to say that the use of coal in the Middle Ages was limited to England. In 1473, university graduate and physician Ulrich Ellenbog of Augsburg was not happy with the use of coal in his city's forges:
To the skillful, subtle, and noble craft of Goldsmithery:
I...have observed the great, severe, and remarkable harm to which this skillful craft comes through the fumes and vapors of the things which the said craft has to work with, so I could not withhold or refrain from giving advice...
In your subtle craft, you use coals for the fire. The smoke or vapor of the same is poisonous...for this vapor burdens the head and also the chest, and when one dallies there too long, the sight grows dim, so that is seems as if there is green and blue and such before one's eyes, or flies in the air. This vapor also makes for heavy, unnatural sleep.
As u/sunagainstgold mentioned, coal was used well before the start of the industrial revolution. They already covered how it was mined, so I'll focus on what it was used for.
As early as the 1300s, coal was used in energy intensive tasks such as making salt, glass, lime, brewing, and to some extent in the production of iron. There was a large increase in the amount of coal consumed in the 16th and 17th centuries, largely driven by the use of coal to heat houses, which by the start of the 18th century consumed perhaps half or more of all coal used in England. It also expanded to other industries such as alum, baking, and the production of dyes.
That heating was not a major consumer of coal in the 1300s, despite it being a large source of energy use, seems to be due to the difficulty of adapting houses to burning coal as well as urban centers still being relatively small-it was easier to source fire wood in rural areas. Burning coal required keeping the fire at a constant temperature that was higher than the average wood fire, and produced toxic fumes that traditional British housing, with a large hall and a fire in the center of, were badly equipped to handle. Modifying houses-adding chimneys, creating covered fireplaces with a grate below to pull air through the fire, and so on, was a long process that required many small innovations by architects and construction workers.
The growth of the city of London in the 16th and 17th centuries provided both of these factors. The size of the city created demand for fuel which outstripped what local forests could provide, and shipping woods from increasingly distant areas to meet demand increased the price of wood to above that of coal. As a growing city, London also had large numbers of architects and builders, who could gradually improve the ability of new housing to burn coal, and disseminate what worked among each other. From there, the best techniques spread through the rest of the country.
Over this time demand for manufacturing increased as well. The cheaper prices of coal in London encouraged the use of coal in a variety of areas, such as baking. There was a huge number of industries that used coal over this time as well, in the production of alum, paper, dye, and so on, as well as in the smelting of brass and copper. Part of this was driven by disruption of imports-England's main source of alum in the middle ages was mines in the Papal states, which was no longer available after they broke from the Catholic church, and there were disruptions in salt imports from France and Spain due to conflicts with Spain and internal conflicts in France.
This increased in demand, primarily in heating but also in manufacturing led to more than a tenfold increase in coal production from 1560 to 1700 (from Allen).
The further massive growth in coal production in the 18th century was driven both by further urbanization, more of the population working in manufacturing than ever before, as well as further new uses for coal. In terms of urbanization, while the majority of urbanization before this took place in London, other cities in England rapidly grew in the 18th century, and building techniques and coal consumption in heating continued to spread throughout the country.
The most striking new use of coal was using coke iron production, which dramatically reshaped the industry, but it started to be used in steam engines as well. Starting with Derby in 1708 and going through a couple decades of further development in Coalbrookdale, by 1750 the technology was ready. Over the next few decades coal rapidly displaced charcoal in iron production, leading to both cheaper and more iron being produced (from Riden). Coal was also used in steam engines starting with Newcomen's engine for pumping water out of coal mines in 1712, which then spread to coal mines throughout the country. By the end of the century improvement in the size and efficiency of steam engines led to them being adopted, although still only gradually, in manufacturing and other industries, though by 1840 the majority of power in spinning and weaving was still supplied by water.
So even before the use of coal in railway and the dominance of steam engines in manufacturing, coal was widely used, for both heating and a huge range of industries.
Sources:
"The History of the British Coal Industry: Volume 1: Before 1700: Towards the Age of Coal", by Hatcher has very detailed descriptions of the variety of industries that consumed coal. There is a second volume covering 1700-1830, but that seems to be impossible to find now? If you want to read in detail which industries used coal and how they used it, this is the book to go for.
Allen's "The British Industrial Revolution in Global Perspective" goes in detail about quite a bit of the processes involved: from how the growth of the city of London drove both demand for coal and supplied an environment where the changes to housing could take place, to the development of coke in iron production and the use of steam engines.
"The Output of the British Iron Industry before 1870" by Riden has detailed statistics on the transition of iron manufacturing to use coke, and on the growth of the industry.
The Cambridge project on the occupational structure of Britain 1379-1911 has estimates on the scale of change of British society over this time, though I'd take the exact numbers with a grain of salt.
Finally, "Urban Growth in Early Modern England: Food, Fuel and Transport" by Wrigely has numbers on British urbanization as well as an overview of fuel usage.