I was under impression that enough was salvageable from the bridge set that they were later able to use it for the Enterprise-D battle bridge and Stargazer bridge - although, it's interesting that the next time we should see this bridge, as the Hathaway, it's a mostly or entirely new set that I think started life as the courtroom in "The Measure of a Man."
As I said, there are two different aspects of a "set" -- the underlying wooden superstructure and the surface facades/dressings. The latter are often far more flimsy and ephemeral than they appear. When we talk about redressing a set, sometimes that means stripping it down to the bare structure and building new facades onto it. That's what they did when they revamped the TMP sets for TNG and then VGR. So it can look completely different to the viewer's eye but still be the same set in that it's still the same underlying framework at the same location within the same soundstage.
And yes, the "Measure" courtroom was another redress of the TMP bridge set. The visible details, the facades and dressings, were different, but they were built within the same framework on Stage 9 and had the same dimensions. Data's cybernetics lab in "The Offspring" and "The Best of Both Worlds" was also a redress of the bridge set.
But of course some redresses are more wholesale than others. For instance, since they made the surface elements for the "Measure" courtroom, it was useful to recycle them for the Hathaway
bridge. Just as the Stargazer
bridge before it had been just a fairly light redress of the battle bridge set, with different consoles and seats but the same shape and structure, essentially just the back half of the set with a new, closer front wall put in.
The thing about the TNG sets being used in TFF that bothers me is that they're so painfully, obviously the TNG sets with no modification. I'm aware that Mr. Shatner liked the aesthetic and hence the bridge was created to match, but TUC proves that some refitting can nicely help distinguish them so it's not so painfully obvious.
That's a surprising comment, because to me it's precisely the other way around. TFF had a bunch of new sets -- the bridge, the maintenance corridors (which were then recycled in TNG: "The Hunted" as the Jefferies tubes), the brig, the turboshaft (much as we try to ignore it), and a hangar deck that was basically a full-scale recreation of the TOS hangar. The transporter alcove is unchanged from TNG, but the rest of the transporter room is redressed back to TMP specs. The officer's lounge was probably a Ten Forward redress, but extensively enough that it isn't obvious. The corridors look the same, but they'd looked pretty much the same since TMP. Sickbay does have some TNG-style Okudagrams on the wall display, but we barely get a glimpse of it. And the engine room isn't seen at all. I'd say that they mostly did a good job hiding the reuse of TNG sets.
Conversely, in TUC, engineering is obviously the TNG engine room with its distinctive warp core and 24th-century Okudagrams. There's a blatant TNG-style replicator alcove in the crew-quarters set. The transporter room has the TFF-style console and shielding, but otherwise it's essentially in its TNG configuration. The officer's lounge is obviously the TNG lounge with its distinctively shaped windows; only the walls and table are redressed, plus the fancy light fixture above the table. The President's office is recognizably a Ten Forward redress, with the same doors and with the shape of the windows concealed only by curtains, while the TFF lounge had different windows and doors. Sickbay has movie-era beds but the overall room is clearly the TNG set. The use of redressed TNG sets is far, far more blatant in TUC.