When Does "Three Men in a Boat" (1889) Take Place?
"Three Men in a Boat (To Say Nothing of the Dog)" is a comedy novel written by Jerome K. Jerome that released August DAY, 1889. So when is "Three Men in a Boat" set?
It takes place roughly in the years:
We know this because the characters get lost in London's Waterloo Station, which opened to the public on July 11th, 1848 AD. So it must be during or after July of 1848 AD. Also, the trip stop in the real-world pub The Crown, which closed in 2008 AD. So it must be during or before 2008 AD.

late 1800s AD
We know this because the characters get lost in London's Waterloo Station, which opened to the public on July 11th, 1848 AD. So it must be during or after July of 1848 AD. Also, the trip stop in the real-world pub The Crown, which closed in 2008 AD. So it must be during or before 2008 AD.
More specifically, the line "some quaint-perched eyrie on the cliffs of Time, from whence the surging waves of the nineteenth century would sound far-off and faint." strongly implies it is currently during "the nineteenth century", i.e. from 1801 AD to 1900 AD. Thus, it's certainly between 1848 AD and 1900 AD.
Finally, the narrator jokes that "I wonder now, supposing Harris, say, turned over a new leaf, and became a great and good man, and got to be Prime Minister, and died, if they would put up signs over the public-houses that he had patronized: 'Harris had a glass of bitter in this house;' 'Harris had two of Scotch cold here in the summer of ’88;' 'Harris was chucked from here in December, 1886.' " The use of "1886" strongly implies Harris is, was, or would be alive in 1886 AD, making any date before 1850 AD or after 1950 AD extremely unlikely as the average human lifespan is well under 100 years.
It has also been reported Jerome based the story on his own experience.
It was adapted to film in 1920, again in 1933, yet again in 1956, a German-language version in 1961, for television in 1975, a Russian-language version for television in 1979, to the stage in 1981, as audio-drama in 1984, to the stage again in 2006, and, again, an audio-drama in 2013.