From what I know of it - based on what my brother told of the archery hobby he picked up for a while - learning to shoot a bow reasonably accurately isn't very difficult or time-consuming. The clincher for military archery was the amount of regular practice needed to develop the physique and stamina necessary to draw a war-strenght bow time and again, which IIRC what I've read of the Mary Rose wreck was in fact enough to cause skeletal deformations in the longbowmen. (This is also another advantage of crossbows, at least when drawn with mechanical assists, and firearms.)

It's worth emphasizing that "yeomen", the class required by the royal proclamations to regularly practice archery, were specifically landowning peasantry - which would make them relatively prosperous in their socioeconomic context. I'm not too familiar with Medieval English rural economics, but at least where I live the equivalent freeholder peasantry quite routinely had varying numbers of paid labourers in their household and depending on the extent of their holdings might well rent out parts of it to assorted landless small farmers.
All of which would suggest the ability to spare comparatively large amounts of time from agrarian pursuits to weapons training - which is why comparable "rural middle class" has pretty universally been the backbone of decent peasant militias. As AFAIK the Crown also saw to the supply of the actual war-bows (in no small part due to the need to import the yew for them and the rather specialised nature of the craftsmanship involved, given the lack of hunting opportunities for commoners in the realm, and for quality-control reasons) when these men were mustered for service this would have lifted a fair bit of the economic hurdles too...

As for composite bows, ehhhh. Do recall that horn-composite staves became pretty mcuh the "industry standard" for fighting crossbows early into the Crusades (AFAIK nobody really knows if seeing the "eastern" reflex bow up close had anything to do with this) - it was the proliferation of this type that prompted the Pope to try (predictably quite unsuccesfully) banning their use against fellow Christians - and seem to have worked well enough everywhere south of Scandinavia, where the rather nastier winters AFAIK play havoc on the glues. (The later steel-stave crossbow worked just fine, though, and after its introduction remained an ubiquitous peasant hunting arm into the 1800s.) Waterproofing bowstaves isn't AFAIK terribly difficult and was done quite routinely - and it's not like the Eurasian steppes, the very home of the composite bow, are either dry or particularly warm year round...

AFAIK the main reason the composite bow never saw much use in Europe beyond the Mediterranean and the Central European steppe interface zone was actually the availability of suitable horn - or rather lack thereof - combined with the relative unimportance of long-rance archery in the context of the "typical" European battlefield (which tended to be rather cramped; this is also why European cavalry tactics evolved into such extremely linear frontal-assault forms, compared to the more fluid and maneuverable Eurasian norm). Plus, since the main point of composite bows is to add draw weight without the kind of unwieldy added stave lenght you get with stronger self-bows, which is mostly of concern for horse-archers (and before them chariot warriors) who as we know were an irrelevant concern in most European warfare (again to no small degree due to geographical reasons), it's hard to see what incentive European bowyers would have had to adopt such a complex, intricate and duly price-hiking construction patterns.
Doubly so as for the most part there was no shortage of suitable woods, unlike in the comparatively treeless steppe and arid Mediterranean/Middle Eastern latitudes.