There are several on-line articles that are a fascinating read but this (themcs.org/weaponry/crossbows/crossbows.htm ) by the Medieval Combat Society is a real eye-opener when you read the timeline at the end of the page, here is one entry:
"1321 Marino Sanuto a Ventian known as Torsello delivered a list of weapons for a proposed crusade and lists crossbows with wooden bows two-foot stirrups, and that composite bows were better in dry areas than in countries with humid climates, M. Jahns, G. Kohler."
or
"1344 and 1366 Dover inventories show 126 crossbows, 34 of which had composite bows with two-foot stirrups and 9 composite bows with one-foot stirrup, and 3 large windlass crossbows, G. Kohler."
That's only 43 of the 126 crossbows that were composite.
or
"1362 Burgundian accounts list 189 light and heavy crossbows with composite bows, and 382 light crossbows with wooden bows. Additionally crossbows with one-foot stirrups, two-foot stirrups and windlasses are listed, and rampart crossbows, B. Rathgen."
There are several yew crossbows still surviving, and, at least one identified as red oak, but I suspect is more likely to be yew. Also, I suspect that a bow needing a two-foot stirrup might be pretty potent when you consider how physically strong fighting men of that period needed to be. Wooden bows are certainly easier and cheaper than metal or composite but in a cold wet climate might also perform nearly as well? For the person paying, easier, quicker, cheaper might be motive enough to ignore 'not quite as good'
