I checked my 1938 Remington ammunition catalog. There is no mention in it anywhere about Remington's using nickel-plated cases in any caliber including .22, .38/44, .38 Super, .38 Special, and .357. There are some color pictures of various cartridges, but none show nickeled cases. Of course, that's not evidence that Remington did not supply nickeled cases, but if so, they didn't advertise the fact.
Nickel plating indeed extends far back into the 19th century, no question about that. But its initial use for cartridge cases is murky. For sure, it was used in the 1950s. I had also heard (without substantiation) that the original intent of nickel plating of cases was to prevent brass corrosion when cartridges were carried in leather belt loops. If that is true, then .38 Special cartridges would probably have been first with plating, as most cops in the early 20th century carried their spare .38 Special cartridges in looped leather belts.
I did notice something else unrelated from that 1938 catalog - Remington was still listing black powder loads for .38 Special. I knew .38 Special was available loaded with BP until the 1930s, but I didn't know it was that late into the 1930s.
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I went back and checked the 1948 Remington ammunition catalog. It DOES specifically mention that nickeled cases were available for .22 RF and also .38 Special. There may be other calibers also available with nickeled cases, but if so, no specific mention was made of them. My guess is that it's likely that case nickeling became more or less widespread during the post-WWII period.