Rather than look at it from the viewpoint of why wasn't something used, look at it from why something was.

The prevalence of horse archery mostly comes from steppe nomads - Huns, Mongols, Scythians, Turkics, and so on. Neighbouring cultures adopted their methods, or were conquered by them. A steppe, in case you can't visualise it, is flat grasslands. Perfectly flat, the biggest stretching from Eastern Europe to northern China. Riding and archery was an essential part of life and needed for survival - the basic necessities of gathering food, as well as raiding and warfare.

The short explanation is that horse archery is the combination of the right materials for the right bows, the right environmental conditions that required the nomadic tribes to use the right horses for battle in the right terrain. It isn't difficult to see how the Mongols could sweep through Central Asia to Eastern Europe.

Look at Britain in the same time period, and you can perhaps see why horse archery wasn't needed. The terrain is much different, the horses are different, and the logistics of armies were different. Horses and mounted soldiers were expensive to maintain, with only the elite fighting on horseback. Archers were a specialised fighting unit, using the materials they had to make simpler self-bows that could be used effectively on foot.

Asking why the English did not adopt horse archery is similar to asking why the ancient Greeks did not adopt horse archery, or the Romans (until much later). The style of warfare was effective in the environment and circumstances in which they were used. The Romans are a key example in that they did adopt horse archery into their armies when they began to encounter enemies that used them effectively in the east.