Every time that I watched STV, just from the dialog and the way the story went, I always assumed that five followed four by only a week or two (I just checked J.M. Dillard's novelization of STV, and on pages 20-22 she seems to have set Kirk, Spock and McCoy's vacation about a few days after the Federation Council's decision and the trial run of the E-A as seen in STIV).
I think I remember this from a few years back when I re-read that one; the onscreen dialogue allows for that interpretation (particularly Scotty's up in Spacedock), though this doesn't account for certain things like
The Voyage Home bridge suddenly getting completely refitted and changed in such a brief span of time, etc. (the concept of removable bridge-modules notwithstanding).
The Dillard novelizations use all sorts of...interesting...dating assumptions, such as the TUC adaptation positing a
ten-year survey mission for Captain Sulu and the
Excelsior having recently concluded in the Reydovan Sector when Praxis explodes. And so on.
(Not unlike A.C. Crispin's
Sarek, which puts a mere three years between TFF and TUC...)
That said, year-long shakedown cruises are a perfectly-canonical event in the
Star Trek universe, as we see at the very beginning of
ST: First Contact, and the NCC-1701-E's first year in space.
I've always chosen to interpret the bridge-chaos as seen in the opening of
The Final Frontier to be the last phases of the overhaul/replacement of the "old" bridge from TVH after that year-long first cruise, with Scotty's men fitting in the last console components when the Nimbus III mission launches.