As I recall:
The original TOS movie bridge, or the rear third of it anyway, was cannibalized into the original E-D battle bridge. This was used mostly intact as the Stargazer bridge and (flipped around) as the Lantree bridge, Wesley's team lab (from "Pen Pals") and the judicial chambers of Starbase 173.
By that point additional bits had been added to it (specifically large trapezoidal shapes above the console that extended the height of the set without needing a full ceiling for it, thus allowing some more camera angles) and were used for another regular guest set which ended up being the bridges of the Hathaway, Enterprise-C, Pasteur and the second E-D battle bridge (which re-incorporated some elements of the older set, notably the four TV screen monitor thingies behind the CO chair). It was used for several guest sets, alien labs, etc. as well.
For DS9, bits of the TOS movie set from TFF and TUC (specifcally the aft third - see a pattern here?) made its way into the Saratoga, Norkova, Prometheus and Odyssey bridges. Dunno what happened after that but the space the set occupied eventually went to the Defiant sets.
The Excelsior bridge set from VOY "Flashback" was used for the Prometheus and Equinox bridges as well. It may have incorporated the same aft third that was previously taken for DS9.
The Defiant bridge was later seen as the bridge of the fake Delta Flyer in "Live Fast and Prosper" and a Ferengi ship later on after that. I think it showed up in ENT as well.
As for Engineering, the TNG Engineering set was the original TOS set anyway, but they only very thiny disguised it for TUC, essentially showing the E-A as having a warp core from a much larger ship 70 years in the future. In TFF they partly redressed the Jeffries tube set they built for that movie as Engineering (in the script anyway), but it still looked like Scotty was just standing in the Jeffries Tube set with a couple generic consoles added.
More obviously, the TOS movie corridors which were converted to the TNG corridors were used in TFF with barely any redressing AT ALL (I think they just left the hallway doors open in a couple shots). THey put a lot more effort into it for TUC, adding lots of extra piping and doors and stuff while still leaving the set basically intact behind it.
Finally, the transporter set from the TOS movies is the same set, modified into the TNG version and then back again - leaving the transporter PADS as TNG era but adding back the glass-paneled booth from the earlier incarnation for TUC anyway. This then become the VOY transporter room with another booth.