The short answer to your question, at least in the west (and particularly the armies of Western Europe) would be the turn of the 16th and 17th centuries. Although there is not exact date, the change was, as you said quite gradual. The main reason that earlier firearms, notably the later version of arquebus, were inaccurate, slow to reload, and for both reasons, really only suitable in large numbers. As such, in many armies, these missile troops were supplemented with more traditionally-equipped weapons well into the later 1500s, such as crossbowmen or in English armies often bowmen. The wreck of the Mary Rose (which sank in 1545) contained both longbows and firearms.
By the 17th century, so the time of the British Civil Wars (1639-1652), the predominant missile weapon of the armies was the musket, although there are some records of highland clan levies in Scottish service being equipped with bows initially, although this was at this point both rare and intermittent, and they were equipped with muskets where possible. Support for non-firearm missile troops remained in society, as there were plans in place to equip pikemen of the period with light bows to give them some limited projectile capability.
The main reason why firearms replaced bows and crossbows in Europe, and this is tied into the wider history of the period, was the rapid growth of army size (overall). It took far less time to train a new recruit in how to use a firearm effectively, even when including the convoluted manoeuvring drill of the period, than it did to use a bow or a crossbow. England's long history of the bow in warfare was based on the medieval laws of regular bow practice over a lifetime, but these had lapsed in the early modern period, making finding and securing practiced, skilled archers rather difficult and expensive to find. The quality of recruit as well was a factor, with many soldiers of the period coming from the poorest elements of the society, who could only be equipped for war at the state's expense and lacked any means of owning (either fiscally or legally) and practicing with weaponry before their enlistment. The growth of army size, even coupled with the parallel growth of state administration and finance, meant that every peace also saw a dramatic downsizing of the armed forces, which would then be rapidly expanded and recruited at the outbreak of the next war, again supporting the need for large numbers of relatively inexpensive but missile-equipped troops.

This period of change in military tactics, equippage, and training, was subject to intense historical debate amongst military historians (and others) during the 20th century. Titled "The Military Revolution Debate" it is still ongoing, but examples of introductory books to the topic include:
Clifford J Rogers, The Military Revolution Debate Readings On The Military Transformation Of Early Modern Europe, Routledge: 1995
Geoffrey Parker, The Military Revolution: Military Innovation and the Rise of the West, 1500–1800, Cambridge University Press: 1996
Jeremy Black, European Warfare, 1494-1660, Routledge:2005