No-one knows for sure, but one possibility is that having lots of trained longbowmen around the place in peace time was a costly risk that a king would need to manage, and only England had the social and political institutions necessary to do this.
This theory was advanced in a paper by the economists Peter Leeson and Douglas Allen (link). Their argument goes as follows:
The longbow was highly effective in the 100 Years' War, and was known to be so by the French and Scots.
Longbowmen take a long time to train and this requires a conscious and sustained programme of laws and institution-building: young men must be made to practice archery instead of doing. But over a 100+-year period, the French or Scots could have made this investment too. (Indeed, both kingdoms experimented with longbows, but did not adopt them en masse.)
But there is another cost to a king of training lots of longbowmen among his people - lots of heavily armed, highly trained, non-noble men milling about the place. These could form the backbone of a rebellion or a revolt. Armed nobles or mercenary crossbowmen were arguably easier to control. (American readers may find the words "second amendment" springing to mind...)
Leeson and Allen argue that England was able to put in place the social mechanisms to control a class of yeoman archers, whereas Scotland and France weren't. (Scotland was poor and often divided; France had its own political problems at the time.)
This might in part explain the tremendous pressure on successive kings of England to invade France - it was perhaps the most important thing in the unwritten job description of an English king in the C14th and C15th.
TL;DR: Once you've trained an army of yeoman archers, you need to control them in peacetime. Only England was up to this socio-political challenge.