This cried out for a test. Fired up a late-game autosave from culture game I recently finished. I had only founded 3 cities, no puppets, full Tradition. Located a 2 pop target (Mayan city, recently founded) to conquer. Spy confirmed the city only had walls, a library and a university. When conquered, the "View City" screen confirmed that the walls and library were destroyed, leaving only a university.
When puppeted (city still in revolt): Nothing but a university.
Puppeted (no longer in revolt): Nothing but a university
Annexed (no courthouse): Nothing but a university
Annexed (after courthouse): Nothing but a university and a courthouse.
Still true 5 turns later (just in case there might be a delayed effect of some sort).
Then I bought a settler and planted a new city: Free aqueduct and free monument
Conclusion: Legalism and the Tradition finisher appear to apply only to the first 4 cities you found. Puppeted and annexed cities do not get those free buildings.