You know, my first thought when I saw this question was, gee, I'd love to have a chart giving the answer for this every armory and maker in the world. However, we can pretty easily see the basic trends.

First, you would find in the pre-1780 world a lot of wrought iron being used for guns. Wrought iron has little or no carbon, and has varying amounts of slag , silica inclusions, running through it somewhat like the grain in wood. How good the iron was depended very much on where it came from, what quality ore and how it was made. You can see very fine hunting rifles circa 1750 that are made with iron that has a lot of slag in it. But some makers took enormous pains: the Spanish seem to have had a real obsession with refining. There are stories of Toledo smiths buying old horseshoes for making their gun barrels, because they'd already been cold worked on the roads. That may not be true, but certainly you find Spanish barrels quite prized, bought and used by other gunsmiths outside of Spain. Muskets of the period would also mostly be made of wrought iron. Even much of the lock would be iron- though, those parts would be case-hardened to make them harder and stronger.

But even then there would be some steel components in a musket- what we would call high-carbon steel. Springs, certainly: maybe all or at least the face of the frizzen if it was a flintlock, and also the ramrod. And sometimes fine hunting guns were steel-mounted: all the little pieces like the buttplate, triggerguard, would be made of steel as well- a pretty expensive option, compared to easier-working brass, And sometimes there would be even more steel parts: breechloaders existed, but required lots of hand-fitting for precision, and some parts for these would be made of steel.

After cast steel and crucible steel (which meant high-carbon) became more common in the later 1700's, you would expect guns to be made with it. But it didn't happen quickly. One of the first would seem to be W. Greener, in England, who began using steel barrels in the 1830's. In the US, Christian Sharps would specify in 1848 that his breechloading guns had cast steel barrels. The Remington company would manufacture the Model 1863 "Harper's Ferry" rifled muskets with blued steel barrels.

But iron was not displaced. The Springfield Armory would make its muskets with wrought iron, and even when it converted them to breechloaders in 1866, and had to sleeve the 58 caliber muskets to .50, the barrel liners were specified as either iron or steel. Only in 1873 would they move to making steel barrels for their guns. And W.W. Greener ( son of W. Greener) would be one of many makers of fine shotguns who would have pattern-welded barrels, AKA Damascus, that would be woven weldments of iron and steel with a thin steel liner.

The steel that was being used for barrels after 1873 was something closer to the typical steel of today, mild steel, with perhaps .5% carbon. That 's what I-beams and re-bar and all sorts of other steel things are made from today, and the advent of structural steel had much to do with the greater economies of scale of the Bessemer converter. Did the Bessemer process greatly affect the small arms industry? It must have. Makers had begun using steel before Bessemer, but wrought iron in comparison became then more expensive and scarce. However, it must have had a bigger impact on artillery, which was commonly made of more brittle cast iron. In the mid 1800's armories began to shrink steel reinforcing tubes around the breeches of cannon- which can be seen on the distinctive Parrot guns of many Civil War battlefields. Soon, makers like Krupps would work out how to make all-steel cannon, and making very big cannon required massive amounts of it.

Around 1890, after the French came up with a workable recipe for a nitrocellulose propellant, steel -really strong steel- was required. It's not hard for wrought iron to withstand pressures of 10 to 20,000 pounds/square inch. But the new smokeless powder cartridges could create breech pressures of 40-60,000. Mild steel yields at around 40,000, and suddenly gun makers had to think about not just mild steel but more exotic steel alloys, and heat-treating, in order to have something that didn't blow up. But even then: given the low pressure of shotguns, the barrel makers of Liege, Belgium would continue making pattern-welded barrels into the 20th c. . The actions of those guns would be, however, steel.

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WW Greener later wrote a history of gunmaking, and though it's quite obsolete on the subject of very early firearms, it's still useful for the 19th c. , and it's over at the Internet Archive.