Depends. Onscreen, we can only establish that TMP can't have taken place earlier than late 2272 (because Kirk hasn't been logging star hours for 2.5 years, and Kirk's five-year mission ended during the year 2270 as per VOY dialogue). We can't establish the last
possible date for TMP very well, except of course as "before TWoK".
We have indirect evidence, but not much of it. At least some starships adopted the new red wraparound uniform jackets by 2278, as per TNG "Cause and Effect", so it would be odd if Earth and the Enterprise
(both central to the Federation and thus likely to receive hot new fashion very fast) continued to wear the TMP wardrobe after that date. However, TMP dialogue also says Voyager Six disappeared more than 300 years before the events of the movie, and in the real world the first Voyager was launched no earlier than 1977.
Our best bet might be to set TMP at early 2278, then, meaning there's a six-year gap to TWoK (in which a bottle from 2283 is considered old) at a minimum. If TWoK and its two sequels take place within three months, as the dialogue rather heavily suggests, and TFF follows immediately after TVH, then there's possibly a decade there until TUC - twice as long as the shortest supportable time between TMP and TWoK.
These are extreme figures, of course, and the truth probably lies somewhere in between - TMP in 2276, say, and TWoK in 2285. There's no way to be more accurate than that.