I am curious to know if it was ever directly stated, or if it can be indirectly inferred as to what year the HMS Bounty winds up in in 'the late 20th century'?
With a bit of hand-waving the date can be indirectly inferred to within a few years using a technology reference from the film and known history.
The use of a Macintosh Plus computer at the plexiglass plant sets the arrival of Bounty no earlier than 1986, since that particular Mac was introduced in January of that year.
According to Wikipedia, hunting of humpback whales was outlawed in 1966 and a moratorium on commercial whaling of all whale species went into effect around 1986, both under the auspices of the International Whaling Commission. Although some countries chafed against the restrictions, the commercial whalers depicted toward the end of the film probably would not have been in operation much past 1986.
The movie description says 1986, but I'm not certain if a character ever actually states the year in the movie. Also according to the IMDB trivia page theres a man reading a may 1986 issue of a magazine in the film.
In the bus scene, there is a man in a brown jacket sitting just in front of the "loud punk." He can be seen "reading" the latest issue of 'Omni' Magazine, which from 1978 to 1998 published articles on scientific developments as well as short works of science fiction. The specific issue in this scene is from May 1986; the cover celebrates the "25th Anniversary of American Manned Spaceflight." (http://graphic-server.com/cgi-bin/usedmagazines.cgi?full/OMNI198605.JPG)