What was the transition of larger militaries from traditional weapons to firearms like?

This is a very good question, and England in particular provides a fairly unique example in its attempts to combine forces using both traditional and modern arms.

In a broader sense, the transition from traditional to gunpowder-based weapons occurred over the course of centuries, but for brevity's sake I'm going to focus on the rise of Pike and Shot tactics in England during the latter half of 16th century.

I've written before about the rise of offensive pike squares that began with the Swiss and the decline of troops armed with “short weapons” such as the halberd, bill, and sword&shield. I've also written about some of the reasons firearms took so long to replace melee weapons completely. Though the proportion of firearms did tend to gradually increase over time, there are examples where larger numbers of firearms were fielded before tactics and technology seem to have caught up, most notably the defeat of Charles the Bold's handgunners by Swiss pikemen during the late 15th century and the disasters which befell French protestant infantry due to their over-reliance on harquebusiers during the French Wars of Religion. Conversely, there were certain situations, such as sieges and skirmishes, in which firearms were more advantageous than others. In the English colonies of the early 17th century for example, pikes and halberds were found relatively useless due to the terrain and small-scale skirmish warfare conducted by the natives and so we see examples of New England militias equipped entirely with firearms well before using the bayonet and around 80 years before pikemen fell out of use in Europe.

Among major powers on the continent, the adoption of pike and shot largely occurred during the Italian Wars. During the second half of the century, William Garrard explained that “Archers amongest [foreign] Nations be neuer vsed, and the Halberd but either amongst fewe or fewe in number.” England on the other hand, though they had been an early adopter of gunpowder artillery, had fielded practically no native arquebusiers until at least the 1530s, and even during Henry VIII’s 1544 invasion of France, the vast majority of English troops were still simple county levies armed with their traditional English longbows and billhooks. Longbowmen would remain in use until the 1590s. The bow remained a very cheap weapon for the average person and in theory every Englishman owned one while for many young levies an arquebus or pikeman’s corslet would have to be provided by a generous gentleman. In addition the use of a pike or matchlock required a great deal of drill to use effectively. This was especially problematic for firearms, which required purchasing a good deal of powder just to learn how to use the weapon effectively.

To make up for this, Henry VIII’s army adopted a sort of composite nature, using native forces to supplement foreign mercenaries experienced with arquebuses and pikes as well as offering higher wages for Englishmen willing to adopt the new weapons. In theory, shorter weapons like the bill were still useful since they were more versatile on rough terrain or in a disordered formation, and they did achieve an enormous victory over the Scottish pikemen at Flodden in 1513. However this seems to have been in part due to the scots own inexperience with continental pike tactics and over time the English seem to have come to agree that armored pikemen in a well-ordered formation were overall much stronger against both infantry and cavalry. In Thomas Digges’ 1579 manual he gives examples of how bill-heavy “battailes” had been arranged in the past, with 7 ranks of armored pikemen in the front to bear the brunt of the fighting with the many ranks armed with miscellaneous short weapons behind to add mass or exploit breaks in the enemy’s formation. Overall he concludes though that “. . . wheras it hath bene among our men of warre a custom to have the bodie of our battails, halberdes, billes, battleaxes, or such short weapons, and to impale them only with certaine ranckes of pikes, I think that custome first grew for want of sufficient numbers of pikes to make a solide battaile.” And that such short weapons are “. . . fitter to execute in a Chase, after the battail is wone, then to decide the furie of a battaile, or abide the charge of Barbed Lances.” Over time, the proportion of bills or halberds in a pike square would grow smaller and smaller, like in continental armies.

As for the longbow by the end of the 16th century they were widely considered obsolete. However, for a long a number time military writers did see one advantage the longbow potentially still held, it could be shot in an arc while musketeers could only fire with one or two ranks at a time. Digges explains that even with complicated countermarch drills, it would be impossible for 10,000 musketeers to complete a single volley in less than 15 minutes, while 10,000 archers firing at once could easily deliver 10 volleys in the same amount of time. On the subject of pike squares he writes that “So that if I should place any weapon within the bodie of my battaile but Pikes onely, it should be long bowes, who may indeed when the Pikes are couched, play over their heads upon their enimies faces w’out any danger to their fellows.” In Thomas Styward’s 1581 treatise he includes a couple examples for how to form a small regiment of 200 men if it includes archers which largely follow the same idea, with the archers usually positioned in deep sleeves or so that they can shoot over the heads of the calivermen or pikemen. When an army was mustered at Tilbury in 1588 to respond to the threat of the Spanish armada, Sir John Smythe observed the placement of the English longbowmen:

. . . they placed certen ranckes of Archers in the middest of their squadrons of Piques, behind the En∣signes, & seuen rancks of Archers they placed behind vpon the verie backe of the battle, and all the rest they reduced into sleeues, close by the flanks of their three battailes, of which sleeues some of them were of fiue in a rancke, and some three in a rancke; and because they should bee surelie garded with shot, they reduced sleeues, or rather squadrons of Caliuer shot close to the flanckes of the Archers, of which sleeues of Cali∣uer shot, some were of nine and twentie in a rancke, other of fifteene in a rancke, and the smallest sleeues of eleuen in a rancke. . .

Ultimately though, this still doesn’t seem to have been that effective compared to fully modernized armies. Sir Roger Williams concluded from his experiences that even in a battle array there was no comparison between archers and shot. Smythe concluded that this was a flaw with the formation itself, “by their fond and vn∣skilfull placing of the Archers, had taken away the whole effect of the volees of their arrowes.”

. . . as they were placed, their sights had not onlie been taken away vp∣pon such an action with the smoake of the shot, and with so manie ranckes and Ensignes closed in frunt and flanckes as were before them, but also the most of their volees of arrowes should haue flien through the taffaties of the Ensignes, and haue glaunced or lighted vpon the piques, either cleauing them, or bea∣ting them downe: besides that, (to the Archers great disaduātage) they should haue lost a great part of their ground, in giuing their volees of arrowes at their E∣nemies, by reason of the distance, so manie ranckes of other weapons being before them:

He concluded that for maximum effect, archers should only be placed at the front and wings of an army, and no more than 7 or 8 ranks deep, as was “The Ancient order of reducing Archers into forme by our most skillfull and warlike ancestours.” Among almost everyone else however, the opinion was that these positions needed to be reserved for arquebuses and muskets, which were much more powerful. In addition the longbow was fairly difficult to use in dense formations in the first place. Robert Barret heavily criticized the practice of lining pike squares with archers, calling the longbow “a cumbersome trying weapon in a throng of men. . . in that place how can bowes, to which it requireth such elbow rome, and are so troublesome, be any wayes in that sort commodious or to be allowed?” In a short time, the longbow would fall out of use completely.

To Conclude: yes, “medieval” weapons were used alongside firearms and pikes for a period of time In England. Bills could be used for fighting on rough terrain or to add mass to pike squares. Longbows could be used to shoot over the heads of their fellow soldiers. However it is debatable whether the addition of these sorts of troops really provided much of an advantage over well trained pike and shot alone.