A quick look at the "usage over time" Ngram graphs on Google for the words 'nineties', 'eighties', 'seventies' and 'sixties' shows that, in print at least, the usage for each was flatlined prior to the late nineteenth century, and that usage grew dramatically in the first half of the twentieth century. Edit - Link to graph.

Also possibly instructive is the existence of this book. The first version was titled "Naples in 1888", and was published in 1888. The updated version from 1897 used the term "nineties" instead, possibly indicating a significant increase in usage in the intervening decade.

Pretty early on after the 1890s, it began to be referred to as "the gay nineties".

They referred to it as the nineties early on. I remember reading a letter by Teddy Roosevelt referring to the decade as such. I believe the letter was written around 1910.

looking through the odd (oed.com) it seems clear it's a 19th century innovation or at least popularized in the 19th century especially with the 1890s.

The 'hungry forties' springs to mind which I'm pretty sure was used contemporaneously.

there is a Disney Mickie and Minnie Mouse Film from 1941 called The Nifty Nineties

There's also a Mae West film, Belle of the Nineties, released in 1934 and set in the 1890s. The Shirley Temple film The Little Colonel (released 1935) is set in the Reconstruction era after the Civil War and opens with a title card reading, "Kentucky in the '70s".