Can I start the National College with all my cities having libraries then settle a third city?

Solution 1:

(tl;dr version in bold)

There are many national wonders that require a certain building in all of your non-puppet cities, the National College being one of them.

If you complete the wonder and only then settle a new city, there is no problem - the wonder is still there even if the new city is missing the required building.

If you are in the middle of the wonder construction, the construction will halt, and you will be forced to choose a different production project. However, the wonder construction will retain its progress, so once the new city gets the required building (and that can happen immediately, if you just buy it), you can continue working on it from the point you stopped. Well, technically there is a production "decay" - I believe that if you don't continue working on the wonder for 50 turns, then it starts slowly losing progress. I always manage to get those buildings long before the decay kicks in, though.

As an aside, as far as I know, the production decay is not exclusive to interrupted buildings. If you just switch a construction project by your own volition, it also applies - the progress will remain for 50 turns (10 for units), and then decay.

This is different with interrupted world wonders; you cannot continue working on them, so instead you are reimbursed with gold.