Back to the original question:
If you redo a tank, how soon can you add back the fish, from the point of view of how established is the population of nitrogen cycle bacteria.
Let me answer the question with a question:
If the new set up has lost some or most of its bacteria....
So you want the bacteria to regrow....
Where are you going to keep the fish, anyway? Do you have another fully cycled tank they can go in, and not cause an ammonia spike because of the large increase in fish population? Can't just keep them in a bucket, or a spare tank that is not any more cycled than your new set up. (and probably has less bacteria).
And how were you going to raise more bacteria in the new set up?
Answer:
1) Refill the new set up with water that matches the original water. Same GH, KH, TDS. pH is less important. Do not add pH altering materials JUST to make the pH match.
2) Put the fish back as soon as the tank water has cleared.
3) If there is any doubt about the population of nitrifying bacteria add a source that includes Nitrospira species of bacteria. Read the label and do not waste money on anything else.