Even without the damage from the rain, the bridge set would have been a decade (at least, I'm not sure if any of it was built for Phase 2) old by the time of Star Trek V and had never been intended for long term use...
Actually the reason those sets endured for so long is because they were
built for Phase II
and thus were designed to be sturdy enough to hold up over the multiyear run of a TV series, as opposed to movie sets which are designed to be used for a couple of weeks and then torn down. The bridge, engineering, and other sets were built for P2 in 1978 and then redressed/upgraded for TMP. They were left continuously standing through the movie years, then were remodeled into the TNG sets and used for seven years in that incarnation, then were remodeled again into the Voyager
sets and used for seven more years. By the time VGR ended in 2001, the 23-year-old superstructure of the sets was rotting and they had to be torn down. But at that point they had long since become the oldest continuously standing sets in Hollywood.
What was ruined, then, would've been the surface elements of the bridge only, the walls and consoles and seats and such. The underlying framework of the set endured through TNG, though it seemed to be gone by VGR.