People did use coal earlier, I'd say the increased mining activity came before the deforestation, but it's a slow ramp up over centuries so gets a bit chicken and egg.
Wikipedia has an interesting history that includes Roman usage. Note the final cite, which has the traditional "we ran out of wood" story is cited to a 19th century source.
> After the Romans left Britain, in AD 410, there are few records of coal being used in the country until the end of the 12th century. One that does occur is in the Anglo-Saxon Chronicle for the year 852 when a rent including 12 loads of coal is mentioned.[8] In 1183 a smith was given land for his work, and was required to "raise his own coal"[9]: 171–2 Shortly after the granting of the Magna Carta, in 1215, coal began to be traded in areas of Scotland and the north-east England, where the carboniferous strata were exposed on the sea shore, and thus became known as "sea coal". This commodity, however, was not suitable for use in the type of domestic hearths then in use, and was mainly used by artisans for lime burning, metal working and smelting. As early as 1228, sea coal from the north-east was being taken to London.[10]: 5 During the 13th century, the trading of coal increased across Britain and by the end of the century most of the coalfields in England, Scotland and Wales were being worked on a small scale.[10]: 8 As the use of coal amongst the artisans became more widespread, it became clear that coal smoke was detrimental to health and the increasing pollution in London led to much unrest and agitation. As a result of this, a Royal proclamation was issued in 1306 prohibiting artificers of London from using sea coal in their furnaces and commanding them to return to the traditional fuels of wood and charcoal.[10]: 10 During the first half of the 14th century coal began to be used for domestic heating in coal producing areas of Britain, as improvements were made in the design of domestic hearths.[10]: 13 Edward III was the first king to take an interest in the coal trade of the north east, issuing a number of writs to regulate the trade and allowing the export of coal to Calais.[10]: 15 The demand for coal steadily increased in Britain during the 15th century, but it was still mainly being used in the mining districts, in coastal towns or being exported to continental Europe.[10]: 19 However, by the middle of the 16th century supplies of wood were beginning to fail in Britain and the use of coal as a domestic fuel rapidly expanded.[10]: 22
https://en.wikipedia.org/wiki/History_of_coal_mining
Seems relevant that the coal mining areas worked out the way to use coal more cleanly in home furnaces a couple of centuries before the 'running out of wood' shift was supposed to have happened.
You might be able to trace whether the trees disappeared first in the areas where they had coal mines.
edit: interesting take on this here:
https://www.historyworkshop.org.uk/wood-scarcity/
> In 1611, the agricultural writer Arthur Standish warned, ‘No wood, no kingdom.’ Deforestation, he claimed, threatened to undermine English agriculture, impoverish the poor, and provoke rebellions. In contrast, his contemporary Dudley Digges – a politician and investor in commercial and colonial ventures – took the opposite position. He argued that fears of wood scarcity were unfounded; a ploy by ‘beggars’ dwelling in forests and the greedy, feckless landlords who profited from these desperate tenants, both of whom wished to protect forests from conversion to more profitable uses. A third perspective was offered by the London merchant and deputy treasurer of the Virginia Company, Robert Johnson. Wood scarcity was real and incurable, and the only solution was to exploit abundant woods in the new English colony of Virginia.