I wrote some comments a while back on the causes of the English transition from longbows to firearms here and here. One major reason for the abandonment of longbows is simply that England was involved in less large-scale warfare in the 16th century than in the 15th century. From the beginning of the 1300s to the reign of Henry VIII, there was pretty constant military activity, which made fighting a pretty good, reliable career. A man looking to acquire a ransom or loot a town could buy himself a bow and a sword and head to France. Archers were paid pretty well, about the rate of a skilled craftsman, and could use their skills to get hired as mercenaries overseas if the crown of England wasn't hiring. By the 1500s, the War of the Roses made the Tudors more paranoid about the enormous corps of professional soldiers available for hire by the aristocracy. The Tudors created laws to reduce the size and number of private military retinues. The Tudors also largely avoided major wars in order to reduce the national budget. Both of those trends meant that soldiering became a less viable profession for the population at large. This is one of the reasons we see fewer and fewer people reporting for militia duty in the middle of the 16h century (no matter what weapon they used).

People were still practicing with bows well into the 1500s, but the pool of militarily skilled archers was simply not there the way it had been in the 14th or 15th centuries. With no real employment opportunities, what would the incentive to practice that much be? There was also a growing consensus among actual military men that the longbow's heyday had passed. Men who had experience fighting in the Netherlands for/against the Spanish believed that the way of the future was a full-on transition to pike and shot tactics. The main group arguing for the continued use of the longbow were not military men but moralists and writers enamored with its role in the English national mythology. Early firearms may have been slow and inaccurate, but they were also better at penetrating armor. With a smaller and smaller pool of archers, the market for archery equipment dried up as well, which increased the price of archery equipment. Of course, that limited the number of people capable of participating in archery even more. Commanders bringing archers to the continent complained that they could not find local stocks of arrows or bowstrings easily. By the 1590s, reality and the military men won out. English armies formally abandoned the longbow, although they could probably still be found in a militia muster for years afterwards.

Just as a side note, what is this if they have a longbow the British would have won the War of American Independence? It reeks of American exceptionalism in the form of 'those stupid british just stood in lines all day and let themselves get shot at' mythology. I'd really like to address that claim, wherever you got it from.