I've had conquered provinces of enemies producing anti-tank, militia and other good stuff that I hadn't researched continue on with their production after capture. One side note, in these situations I never had to repair any buildings. I don't know if the production would have continued if I had to repair some building first.
To produce a cruiser a level 2 naval base and level 1 industrial complex should be sufficiënt, so you only need to repair the industrial complex to level 1.