Judging from what I've seen in my collection of American Rifleman magazines dating back to 1928, nickel cases appear to have been introduced in the 1920s.
Anyway, there seems to be a number of announcements about the new, nickel cases. Or maybe it was advertising geeks using the adjective "new" to renew interest in an already old item.
It's a good question and one I've not seen addressed anywhere. I wonder who made the first nickel-plated cases in general production?
I'm sure that cases were plated by amateurs or in test runs shortly after fixed ammunition was introduced.
I recall that the U.S. military produced a batch of --- .30-40 or .30-06 ammo, can't remember which --- with tin-plated cases. This proved to be a dangerous mistake.
After a few years, the tin naturally bonded itself to the jacketed bullet. Shortly after production, it was noticed that many service rifles blew up catastrophically.
An investigation uncovered the tin-plated case as the culprit.
I believe there were a few .45-70 cases that were tin plated as well, but they didn't suffer the same problem. Apparently, the tin of the case bonded electrolytically with the nickel or copper of the jacketed bullet, but not with the lead bullet used in the .45-70.
Sorry for speaking so generally, but I'm at work and away from my resources.
Nickel-plated cases became popular generally in the .38 Special, then it was done to other cases. In the 1920s and up to the 1970s, police officers carried their .38 ammo in cartridge loops on their gunbelts. The acids used in the tanning of leather attacked the copper in brass cases, causing green corrosion (verdigris).
If the corrosion was not regularly wiped off, it could get so bad that it was difficult to remove a case from the belt loop, or the corrosion on the case kept the cartridge from being pushed into the revolver's cylinder.
To counteract this corrosion, nickel-plated cases were touted as the cure. To a point, but even nickel-plated cases will begin to turn green wherever the leather contacts the case, given enough time.
Nickel plated cases are still useful. In semi-autos and bolt-action rifles their slicker nickel case tends to feed into the chamber easier than a brass case.
They are also useful for identifying particular loads. In my .44-40 rifle, I use nickel cases for black powder loads only. If it's nickel, it's a black powder load. If it's brass, it's smokeless. Also, black powder fouling cleans off the nickel cases easier.
Nickel-plated pistols (revolvers and single shots) were enormously popular in the Old West. It's thought today that this popularity stemmed from the ease with which black powder fouling could be cleaned off the gun. It's also easier to spot the fouling in the nooks and crannies of a nickel-plated pistol.
