Surely stuff like the chairs, some light fixtures, and other bits of the set dressings would be kept in storage between JJ-verse movies along with all the uniforms. They'd be warehoused somewhere and in this case shipped to Vancouver to fill out the newbuild sets they would have erected for the film. It'd be more cost and time effective to do this than to physically disassemble and ship the sets from LA, and not that big a deal with a bigger budget production.
So for the new series, they'd have access to those bits of set dressing, plus uniforms, but most of it is specific to the look of the JJ-prise and that entire era. If they want to make a break from that, very little would be useful. As for soundstages, it's overall just a big empty room with varying levels of sound insulation and a large lighting grid suspended from the ceiling. Some of the TNG sets that weren't recycled from the TNG sets were built to certain size specifications so they could match bits of the lighting grid and thus make certain lighting situations easier; but more often than not it simply didn't make a difference.
I think the old transporter room set was the most long-lived of them all, being build for TMP and then refit for TNG and then Voyager while pretty much the rest of the sets were bulldozed and equivalent ones built in basically the same place. The TNG bridge was replaced by the VOY bridge, the corridors, sickbay, quarters, and engineering sets, same thing.
Mark