I first noticed certain video bloggers pronouncing button as "BUH-ehn", with a distinct glottal stop between syllables, sounding like an overt attempt to avoid enunciating the "t". While button is the most egregious example, I've heard t's dropped in other words as well. I've even noticed coworkers doing this now, so it seems to be catching on. To what can we attribute this phenomenon?
[Later note]: Maybe it has been around awhile, but maybe I wasn't clear about the exact sound. There is a marked effort to pronounce "ehn". Most people say "BUT'n": no release of the T-stop, with a glottal stop between the t and the n. The pronunciation I "recently" noticed (and am hearing more often) is "BUH-ehn" — no T-stop at all and minor emphasis on "ehn". It almost sounds labored, like extra effort to avoid the T for some reason.
Best Answer
Sometime in the 1500s.
https://www.merriam-webster.com/words-at-play/correct-pronunciation-of-often
But this is probably a different phenomenon than the t-glottalization you're noticing. According to the Wikipedia article, "The earliest mentions of the process are in Scotland during the 19th century" ("T-glottalization"). Most of the written materials I can find focus on British accents, but this paper about t-glottalization in American accents is interesting: "T-GLOTTALIZATION IN AMERICAN ENGLISH"
The authors cite a 2006 study that suggests that t-glottalization is most common among adolescents in America. They also suggest that it's more common on the West Coast, then the East.
Anecdotal
My own experience, having grown up on the West Coast and moving to the East Coast as an adult, actually suggests the opposite.
Actually, the t-glottalization in the Northeast may not be more frequent, but it seems somehow "stronger." That is, I and virtually everyone I know (including my parents, who are in their sixties) pronounce button with a glottal stop, not a /t/. However, I have met a number of twenty-somethings raised in the Northeast who pronounce button differently, as if their tongue tip doesn't touch the roof of the mouth until the n. I believe that when I say "button," my tongue tip rises to make an unreleased t, perhaps simultaneously with the glottal stop.