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Hoo boy. Buckle up.

First, check out these two earlier answers that address parts of your question:

• I am a homosexual man living in the western world during the Victorian or Edwardian period. How do I discreetly meet other men for sexual encounters? by /u/cdesmoulins
• In the 17 and 1800s, how did gay men find other gay men for sex? by me

Three important caveats, before I launch into this screed:

1. For simplicity’s sake, in this answer I’m going to use “gay” as shorthand for the more accurate term “men who have sex with men.” “Men who have sex with men” is a category often used in social science research to refer to men who identify as gay, bi, pan, etc., as well as men who identify as straight but have sex with men, plus men who prefer to avoid labeling their sexual orientation. But it’s a clunky term, so “gay” it is!
2. Since I’m a British historian, I’m going to focus on British experiences; hopefully one of my colleagues will be able to tell us more about gay men in other parts of the world.
3. Factors like religion, class, and geography heavily influence(d) men's daily experiences. Because of the limitations of the tools of history, we have a somewhat skewed view of historical gayness. Middle- and upper-class men had the time, education, and equipment to leave copious written records, while working-class and poor men likely did not. This means that we know a lot more about relatively high-class gayness than low-class gayness, though poor and working-class gay men had vibrant and important cultures and lives. Also, the history of queerness overlaps heavily with the history of cities (see the first part of this answer by /u/xyti099 to learn why), so there is a tendency to overlook rural gay lives--but that doesn't mean that there weren't gay folks in the countryside.

Now onto my real response.

The very end of the 19th century was a weird time to be a gay man in Britain. In one sense, things were bad and getting worse. A series of very public scandals in the 1870s (e.g., the Boulton and Park affair) and 1880s (e.g., the Cleveland Street scandal) had raised awareness among the general public of dissenting practices regarding gender and sexuality. Whereas gay men earlier in the century could often--to a degree--go about their business without the cishets noticing what was happening, by the end of the century Britain was on high alert for nonnormative sexuality. In 1885, the Labouchere Amendment to a broader law aimed at protected women and girls from sexual exploitation made the amorphously defined "gross indecency" between men illegal for the first time. (Previously, penetrative sexual intercourse between men was illegal but hard to prosecute, but Labouchere criminalized a range of behavior like kissing.) The 1895 Oscar Wilde trials had been the pinnacle of this popular awareness of sexual desire between men; in May of that year, newspapers had overflowed with reportage about the so-called telltale signs of gayness--everything from classical allusions in private letters to soiled hotel bedsheets. All of this worry about a perceived decline in the sexual morality of Britain's men was part of a larger panic about degeneration (which is basically the precursor to today's Telegram and talk-radio narrative that Kids These Days are immoral and too sensitive for their own good). The coming Second South African War (1899-1902) would be popularly taken as evidence that British masculinity was in decline.

On the other hand, there was an absolutely thriving scene for gay men, especially those who lived in big cities. London particularly had a vibrant gay culture, complete with established cruising grounds. (See Cook's London and the Culture of Homosexuality for more.) If it seemed like the law was getting too close to comfort, a man with means could always hop across the Channel to France or go further abroad to Italy, Greece, or North Africa, where sex between men (and sex with boys, too--see Boone's The Homoerotics of Orientalism) wasn't criminalized. The ever-growing British Empire, though it would seem to falter briefly during the South African War, offered nearly limitless opportunities for sex with men, if one desired to join up in the army, settle on some Indigenous people's land, or find a nice trading post in some faraway port (see Aldrich's comprehensive and accessible Colonialism and Homosexuality).

The advent of sexology over the previous half century, too, meant that gay men in 1899 were creating and finding new ways to think of themselves. Invert, as you note, was one term for what we'd today call gay or lesbian people. "Homosexual" had first been invented as an adjective in 1869 (as "homosexuelle" in French) by Karl-Maria Kertbeny. Richard von Krafft-Ebing’s Psychopathia sexualis appeared in 1886 and catalogued a range of non-normative sexual behaviors. In 1864, Karl Heinrich Ulrichs proposed the German term "Urning" to refer to men who loved men, and English speakers often used the English version of the word: Uranian. (Of course, this new vocabulary of sexual identity allowed for self-expression, but it also facilitated surveillance.) A lot of sexology happened on the Continent, but interested Britons took note and formed correspondences with the European sexologists as well as each other. Men like John Addington Symonds, George Cecil Ives, and Edward Carpenter were key holders of gay knowledge and crucial axes of gay sociability at the time.