Okay, so over the history of Japan and the bow, a lot of it came from the basic need for archery during the wars in the earlier years, and then there's a lot of popularity that stemmed from the same era that immortalised the katana into Japan's culture.

Also, I'm not too sure how much you mean by ingrained into Japanese culture - it's not really something you'd hear many people of Japan mention when you prompt them with 'Medieval Japanese Weapons', and it certainly isn't the first thing most Western people think of when you say the same - yes, archery (or as some prefer, kyujutsu or kyudo) did have an important role in Japanese warfare, but was it significant to the extent that it is now deeply ingrained in modern Japanese culture? Not too sure about that (I've certainly never heard it mentioned).

During pre-feudal Japan (and really into feudal Japan too) much of Japanese warfare was centered on the horse and bow (think Heian, Kamakura, 12th century and onwards). Similar to bows on the main Asian continent in civilisations such as China, composite bows were favoured and often made from laminated wood or bamboo. Simple bows were easy to draw, allowing for a swift succession of shooting when in relatively close quarters.

These qualities allowed it to transition into a popular tool for hunting, ceremonially and competitively, in a way kickstarting the development of techniques such as kyujutsu – one of the famous schools off the top of my head is ogaswara ryu, a culmination of bajutsu (horsemanship) and kyujutsu (archery) – essentially, mounted archery.

To the second point on popularity – the Edo period. When the Tokugawa shogunate established essentially two and a half centuries worth of peace, along came the idea of recreational techniques in handling weapons, such as the sword and bow. The modern iteration of kyujutsu, kyudo, sprung up around this time, and archery spread from an essentially nobility-exclusive skill to one that was practiced by all castes.

Kyudo really is an umbrella term that encompasses all sorts of techniques, from emphases on flow and form to efficiency or power, and continued onwards (similar to the progression of the katana and its techniques) from the Edo into modern day.

In comparison to England as per your example, I probably can’t shed too much light on a detailed comparison, though in summary for archery in Japan – necessity as well as cultural shifts allowed ‘The Way of the Bow’ to move its way to the top.