I've heard that many English archers were recruited from the peasantry and expected to practice their shooting weekly either at home or in a group setting. I've also assumed that "able bodied" men capable of bending an English warbow appeared in a variety of sizes, but a handful of archery experts on YouTube insist that there were unwritten height standards that the nobility enforced. I know next to nothing on this subject myself; I'm curious what reddit's opinion of this is.
There were no standards, either written or unwritten, for height or size for any English soldiers, whether longbowmen or otherwise. By the last half of the 14th century, English soldiers were increasingly being recruited by contract. The standards established by these contracts were more focused on equipment than on physical standards. An archer recruited for a campaign would have been expected to have the equipment required for his role: i.e., a mounted archer must have a horse, bow, sword, etc. There were no physical standards; a captain looking over his men at a muster would not be measuring them and weighing them. Rather, they were expected to be able to effectively wield their weapons. A one-armed archer who arrived at the muster would not be accepted, but no one would have blinked at a shorter-than-average soldier signing a contract so long as he was capable of fighting. The "nobility" did not have the resources nor the care to enforce some kind of arbitrary height standard. If a prospective soldier signed up, was capable of fighting effectively, and had the equipment to do so, there would likely be a place for him in a company of archers, unless the circumstances were particularly unusual.
/u/MI13 has covered the issue of whether or not there were written or unwritten standards, so I'll talk about the height.
The average height for a medieval man in England was around 5'7" or 5'8" depending on the period and which study you read. If you were to say that there was an average height of an archer, this would likely be it. There are some archaeological finds with probable archers who were taller than this (Towton 16 was 5'9", while the archer the Mary Rose Trust did a facial reconstruction for was also 5'9"), but this probably reflects archaeological bias to some degree. The same size is quite small, and there is sufficient natural variation to account for this. There is possibly a further bias in that some of the skeletal deformations used to identify archers require the use of proportionally heavy bows during the person's youth (such as os acromiale and Medial epicondylar apophysitis) whereas Richard Wadge has examined arrowhead finds in Oxfordshire and found that those not found in areas where professional archers would have been came from relatively weak (<80lb) bows. Any number of men may have chosen to learn how to use heavier bows after they reached adulthood and so the effects on their skeletons are less pronounced.
Further, as hinted at by /u/MI13, it was not uncommon for sons, younger brothers or other family members to accompany a fully equipped man-at-arms on campaign as their archer. These would have been comparatively better fed than most commoners, and are likely to have been practicing with their bows from an early age. These are the people most likely to be identified in the archaeological record as archers, due to their practice at a young age, so their skeletons are likely to skew the height slightly higher than average.
Sources
Blood Red Roses: The Battle of Towton
Medieval Arrows From Oxfordshire
Arrowstorm: The Archer in the Hundred Years War