The funny part is that during the bridge evacuation, the Turbolift that Data was pretending to walk to was the Battle Bridge Express and Picard didn't seem to think anything odd about Data heading to the battle bridge when everyone was ordered to reassemble in engineering.
Also, why should Data evacuate in the first place? He is not reliant on life support, the impending failure of which was the reason for the evacuation!
Considering that the ship was accelerating out of control, moving from one bridge to another would not have been particularly fruitful. Odds would not be good that the secondary bridge would have any more control over the warp engines. Brainstorming at engineering would be the way to save the ship; after that was done, the officers could return to one bridge or another.
Once life support on main bridge started to fail, though, Picard might have suspected specific problems with that facility, in which case the battle bridge might indeed still have been functional and capable of controlling the warp engines. But engineering would still be the generic solution to all the woes.