I'm not a military historian specifically, but I do study the period, so please forgive my broad generalizations.

The adoptions of firearms as a military necessity followed somewhat different paths for cannons as opposed to small arms.

1. As cannons and siege artillery, firearms are introduced to the rest of the world from China in the middle of the 12th century. By the end of the 13th century cannons have become an essential part of siege warfare in many societies across Europe and Asia. So in that sense, firearms changed the ways sieges were fought fairly quickly after their introduction, although this process still took a century or more. Nevertheless, by the end of the 14th century older types of siege weapons like catapults and trebuchets were rarely used except when armies found themselves desperately short of gunpowder during a siege.

By the early 15th century, new types of fortifications, made chiefly of layered and interlocking lines of earthworks, the trace italianne "star fort", are being created to defend against cannon which can easily destroy the thick masonry walls that had previously been effective in the defense of fortresses and cities.

2. In contrast, the adoption of smaller man-portable firearms as a infantry (or cavalry) weapon was a much slower process that took much longer for both technological and social-organizational reasons. I know less about this process, but basically we don't see the large-scale use of firearms by infantry formations until the Italian Wars of the early 1500s, and even then adoption of the arquebus was a gradual process, partially because it nesseciated new types of formations and tactics. The Spanish develop the tercio formation in the 1530s, which sees the tactical integration of pikeman and arqubusiers into a single unit. What's notable here is that arquebuses of similar weight and rate of fire (which is slow, often around one to two shots per minute at most) are extant since the mid-1450s, but it takes several decades for logistical and tactical thinking to figure out how best of use firearms as an infantry weapon on the battlefield.

So in summary, cannon were adopted fairly quick as soon as states and armies possessed technical and economic abilities to field them, because they fit neatly into the role previously held by earlier types of siege weapons, and the weight and slow rate of fire weren't really a problem during a siege, and because they very effective at destroying the older types of masonry fortifications that had been in use the previous 1000+ years. In contrast, the adoption of firearms as an man-portable infantry weapon took a bit longer since it wasn't as immediately obvious was the best way to use them was, and so widespread adoption of small arms required the development of new tactics and new types of military organization to make them useful.