As a loanword from Italian, latte (as in a latte, and distinct from combined forms like caffe latte) is from the US, and likely the Pacific Northwest, circa the mid-to-late 1980s. (It first appears in print by itself in 1987.) This answer will first cover the OED entry, cite other early examples, and then hypothesize based on the history of coffee culture in Seattle in the 1980s that Howard Schultz may have helped popularize the drink if not the term.
Oxford English Dictionary - when latte became mainstream
The OED entry for "latte, n.1" gives its earliest quote from a 3 May 1989 article in the New York Times:
1989 N.Y. Times 3 May c8/4 The most common order heard is for a latté, espresso with milk, another variation on cappuccino and café au lait.
In general, my searches confirm that a latte is in widespread use by the early 1990s, and I find it plausible that the New York Times is one of the first national publications to pick up on the trend.
This is a case where tracing the article to its original source tells far more about the context.
Latte may have emerged on the west coast
The original New York Times article from 1989 ("DE GUSTIBUS; In Seattle, Espresso is Raised to an Art") connects the growth in espresso drinks to the emerging coffee market in Seattle. Two interviewees provide a description of a lively coffee house culture developing in Seattle starting in the 1960s and 1970s:
Kent Bakke, owner of Visions Espresso, an importer and distributor of coffee and espresso equipment, said the quality of the coffee was a key. ''In the late 60's and early 70's we were developing our palates in Seattle,'' he said. ''All things related to food and wine - and coffee is part of it.''
Debra Hackett, who is a sales manager for S.B.C., one of the city's large roasters, says her company and the other big roaster, Starbucks Coffee Company, ''have been responsible for educating the public about quality coffee, to appreciate that it is better to have one good cup than five or six poor ones.''
The reporting indirectly suggests that latte, like other espresso beverages, emerged within the growth of the coffee market in Seattle. My attempts to trace other sources similarly suggest latte is only present in west coast authors and publications but not elsewhere, indicating a possible point of origin on the west coast. Take for example "Great tastes and scents at a no-nonsense price" (July 17, 1987) in The San Diego Union, which reports on a local shop:
There's no better way to greet the day than with a cup of excellent espresso or cafe latte at The Kiosk in Del Mar. This small, unassuming shop features freshly roasted beans, low prices (90 cents for an espresso; 95 cents for a latte), orange juice squeezed while you watch, great chocolate-chunk cookies, breakfast muffins and friendly service.
In the same year, in The Vancouver Sun, writer Eve Johnson reports on a place called Cafe Zen, "A breakfast place to soothe troubled souls" (16 October 1987):
The specials seem to have been constructed on the basis of whimsy: $4.95 buys Eggs Florentine, half a grapefruit and a latte or cappuccino; $2.95 buys a half order of Eggs Florentine (one egg) and regular tea or coffee. You can get latte instead by paying the difference, and on fragile mornings, you certainly should.
Then there is Jan Grover, who worked in San Francisco in the late 1980s, writing about preferred beverages to have while reading M.F.K. Fisher (Grover, J. Z. (1989). The Pleasures of Memory. The Women’s Review of Books, 6(9), 1–4.):
If, like me, you prefer to choose your time and place to eat – if a latte or a chilled Sauternes or a Sierra Nevada on draft in the company of strangers enhances your day, makes your writing more enjoyable, then seek out the company of M.F.K. Fisher.
(Side note: Sierra Nevada Brewing, an early craft beer place, started in California in 1980, so this quote cites multiple relatively trendy beverages.) In other words, the earliest uses for a latte I can track (see note 1 below) all pertain to the west coast. I hypothesize the origin is still more specific: Seattle in the mid-1980s.
Caffe latte and the Seattle coffee culture
First, a little history. In the 1960s and 1970s, Seattle had two kinds of shops related to coffee. The first would be the coffee shop or cafe, a sit-down restaurant focused on broadly American food and offering drip coffee to drink. The menu for Coffee Dan's from the Seattle Public Library, undated, in Seattle is a good example: big focus on breakfast with beverage options (coffee, tea) in small type in the bottom right. These are familiar establishments across the US. The second would be a coffee roaster. Peet's (1966) was an early specialty coffee roaster coming out of Berkeley, CA; Starbucks (1971) and other roasters soon opened up in Seattle (see "Third wave of coffee," Wikipedia). They were focused on selling coffee beans, not brewed coffee. Nonetheless, they raised customer awareness of coffee sources and types.
By the mid-1970s, a new kind of coffee bar began to pop up in Seattle, starting with Cafe Allegro (Wikipedia). Cafe Allegro and similar shops focused more on the types of coffee brewed. In its early menu (via Seattle Public Library) one can see the differentiation of kinds of coffee like espresso, cappuccino, and mocha, albeit no latte or caffe latte:
Based on several fruitless searches, I'm inclined to believe that latte wasn't around at the time. For example, a Seattle Times article by Keming Kuo titled “Coffeehouses: Beating the Rat Race” (February 25, 1978) describes many of these early coffeehouses and details offerings like a 35 cent espresso from Otto’s or a wide selection of teas, coffees, and candies at Yetta’s. Lattes are unmentioned, and I've found no other mentions from that time.
That brings the possible date for latte to the mid-1980s. One clue: in 1986, Caffe latte first appears on the menu of a small coffee bar called Il Giornale. The bar was opened by Howard Schultz in 1985-6 and merged with Starbucks when Schultz bought them out in 1987. The menu (also via SPL) includes the following:
Caffe Latte
(espresso and steamed milk)
How this item got on the menu may be an origin point for latte being used instead of cappuccino or cafe au lait, prior English terms for milk with coffee. Howard Schultz founded this coffee shop after working at Starbucks in marketing for a few years. In 1983, Schultz took a trip to Italy, where he drank his first latte (Sheila Farr, Starbucks: The Early Years). By 1984 Starbucks, which had opened its first coffee bar in 1982, would include the caffe latte on its own menu (Starbucks Company Timeline, via Internet Archive Wayback Machine). Schultz left to found Il Giornale by 1985, where he kept the drink.
Caffe latte may have soon spread to other establishments, and they may have also begun to adapt the term latte to refer to the new drink. By September 1987, the same year as the earliest newspaper articles in Vancouver and San Diego, latte by itself appears on the menu for an unrelated Seattle bar, Gravity Bar (University of Washington Library):
My hypothesis is that latte comes from the early uses of caffe latte in west coast coffee culture centered in Seattle and possibly from Howard Schultz himself, which is why the verifiable print dates for a latte in 1987 come shortly after coffee bars started selling the caffe latte in 1984.
Research process
For this post I consulted several databases in an attempt to find usages of latte prior to 1990. My general search terms included "a latte" (more restrictive) and "latte" (less restrictive, including caffe latte, the last name Latte, Latte stones, (Wikipedia) and so on). I skimmed results to check for pertinence and attempted to verify dates within the source if possible or with corroborating sources. The list includes:
The British National Corpus (1980s-early 1990s) - no results
Google Books - unrelated results
JSTOR - found The Pleasures of Memory (1989); otherwise unrelated results
ProQuest, News and Newspapers - in "Oldest First" search, first eight results in 1980s all tied to west coast publications (Los Angeles Times, Vancouver Sun, San Francisco Chronicle, Seattle Times, Orange County Register); ninth
result is May 1989 NYT article "DE GUSTIBUS." Six remaining 1980s results are west coast.
Gale OneFile - has "DE GUSTIBUS," no other results.
As part of my search strategy, I also found some results tied to library archives. That led me to search for material from digital restaurant menu collections from the Seattle Public Library as well as the University of Washington Libraries, setting aside sources where the date was not verifiable. Since these sources were necessarily location-based, I ran a comparative search with another menu collection (the New York Public Library menu collection); the only valid result was for caffe latte in 1989, Coffee Etc. in Tucson, AZ. Since menus are ephemeral texts, it's possible undigitized archives or unarchived material exists with earlier usage.
None of these searches turned up credible evidence for a latte existing beyond the west coast before 1989. That is one reason why I hypothesize usage to originate in the west coast of the US and Canada.