Paul W. answered 04/05/19
MA Military & Diplomatic History with 10+ Years of Teaching Experience
There were particular locations in Europe during the High and Late Middle Ages where the crossbow became a particularly popular weapon. This, of course, doesn't mean that crossbows weren't used by people in other locations (it was a popular weapon, for instance, in Scandinavia during the Late Middle Ages). It does appear, however, that there was a connection between wealth generated by trade and the degree to which the crossbow was in use.
It should be understood that the crossbow was a relatively expensive weapon in comparison with the other weapons of choice in use by foot soldiers. Crossbows, by Medieval standards, were complex machines. They required a number of different skilled craftsmen to produce and assemble their distinct parts. The wooden stock, in which was contained a mechanism composed of moving metal components, and a compound (later, a steel) bow - all required a particular type of master of the materials involved (I haven't even mentioned the mechanically sophisticated spanning tools used to draw back the bowstring after firing...).
It's no accident that locations such as Flanders and the Italian city-state of Genoa became well known for their crossbowmen. These were centers of trade where existed a combination of concentrated wealth necessary to pay for the manufacture of crossbows and a concentration of craftsmen - members of urban guilds - necessary to accomplish the manufacture of crossbows.
Moreover, cities contained the wealth that allowed both municipal governments to arm urban militiamen with crossbows and / or citizens who themselves could afford to purchase a crossbow. Men armed with these powerful, specialized weapons could demand higher pay for their services than foot soldiers armed with bows, spears, or polearms. The higher wages would, in turn, allow crossbowmen to pay for the repair or replacement of their weapons.
You may be correct that a peasant from Yorkshire could be taught how to correctly use a crossbow. But who would pay to purchase one for him - a peasant was not likely to be able to afford to purchase one for himself (though the wealth of peasants varied considerably). In any case, why go to the trouble when the average Yorkshire peasant already had a perfectly good weapon, the longbow?