So you might be interested in this comment where I replied to a related question from /u/a_tedder

First off it is important to note that archery was not widely introduced with the Norman Conquest. In fact if there was any military technology/tactic introduced by the Normans it was the focus on the use of heavy cavalry (knights/men-at-arms) as opposed to heavy infantry (huscarls/thegns). Archery did play a role at the battle of Hastings (Harold Godwinson was purportedly killed by an arrow in the eye) but these were not what historians consider the longbow. In fact, as I mention in the post linked above, widespread use of the longbow really only caught on during the Plantagenet Kings, specifically Edward I-III. As I detail, the defining peripeteia in terms of the use of the longbow was the conquest of Wales by Edward I in 1283, over two hundred years after the Norman Conquest. Anyway I digress.

In terms of Saxon England archery does not seem to have been mandated in the same way that other forms of fighting were - for instance rules surrounding what a warrior must be equipped with during the saxon period primarily focus on melee combat, and that combat was almost always dismounted. There were few saxon archers at Hastings, for example. The defining characteristic of Saxon warfare seems to have evolved around the shield-wall, which stems from Germanic fighting traditions. Archery may well have been present, but not in enough mass to turn the tide of the battle.

Another case would be Scotland, which evolved as a Kingdom and a defined polity at a similar time as England was being unified. Archery may have been a part of Scottish warfare (though as a general rule there are almost no detailed sources for this kind of thing and when there are they aren't usually focused on the specific weapons the opposing forces are using) but in the battles we have records of they either rely on melee combat (Battle of the Standards) or the use of massed Pikemen (Falkirk, Bannockburn, Dupplin Moor). Actually Robert the Bruce, King of Scotland, actively tried to institute archery training in a similar way to the English after coming to understand its potency, but ultimately failed to change much.

Ultimately the use of the longbow as a primary part of combat seems to have its origins in Southern Wales, where Gerald of Wales would have us believe it was widely used by the native Welsh. These men would become an increasingly important part of Edward I's armies in Wales, Scotland and elsewhere, where they would contribute mightily to English military success. By the time the English mandated through law that normal people practice archery it was already in significant use throughout southern Wales and parts of England, and had shown its worth on the battlefield. The laws you mention were more a reflection of English rulers taking advantage of what was already a growing trend.

Davies, R. R. The Age of Conquest: Wales, 1063-1415. Oxford: Oxford UP, 1992.

Morris, J. E. The Welsh Wars of Edward I: A Contribution to Mediaeval History, Based on Original Documents. London: Oxford UP, 1968.

Prestwich, Michael. Edward I. Berkeley: U of California, 1988.

H. R. Loyn, "Gesiths and Thegns in Anglo-Saxon England from the Seventh to the Tenth Century", The English Historical Review 70, no. 277 (Oct. 1955)

Fleming, Robin. "Rural Elites and Urban Communities in Late-Saxon England." Past & Present, no. 141 (1993)