The mod may be installed before, during, or after the Civil War; it makes no difference.

The quest starts soon after the final battle in Solitude or Windhelm.
You'll be notified (not by the magical courier -- by the starting quest) that city repairs have begun and that in the Temple of the Divines in Solitude is available a donation box. You may offer some gold to speed-up the reconstruction.
All the cities will be repaired: Solitude or Windhelm, and Whiterun.
The fundraising lasts two days; if you don't go to the temple or don't give a donation, the box is removed.
If you offer some gold, the box stays and is locked (is there a key somewhere?... yes) -- after 24 hours the gold will be removed.

Donation effects
No donation -- Repair in about 20/21 days
5,000 -- Repair in about 15/16 days
10,000 -- Repair in about 10/11 days
15,000 -- Repair in about 5/6 days
Steal the donations -- Get back your gold plus some honest citizens' offers and much shame. Repair in about 30/31 days.

A note...
I imagined the weary hero walking to the city gates after the victorious battle and riding away into the sunset upon his noble steed.
This means that the quest only starts when you're not in one of the five hold capitals; if you mainly fast-travel from city to city it might be quite delayed.

And another one...
The quest completes when you enter an interior cell after the required period has expired; this is to prevent the oddity of seeing swapping buildings and disappearing stuff.

Add-ons

• Enjia, an inconsequential tenant living in the repaired Heimskr's shack if and when he's jailed. She just stays around and pays the rent.
• Nimriel inherits the repaired Pelagia's house and sleeps there.

Patches

• The patch for "Immersive Citizens" allows Heimskr to go back to his rebuilt house. It overrides Heimskr's sleep packages with different conditions so that he doesn't spend the night standing where his bedroll is no longer there. All this in case of a Stormcloak victory, otherwise he'll stay in his cozy cell. If you do have Immersive Citizens but don't care about Heimskr's well-being, just don't install the patch.
• The patch for "Open Cities Skyrim" is only for the repair crews, the repairs themselves always work, even without the patch.
• A patch for "YOT - Your Own Thoughts SE" is here, not sure whether it's being kept up to date, though.
• The patch for "The Second Great War" delays the quest start till after the end of the... second great war.
• The patch for "JK's Temple of the Divines" moves the box to the repositioned cabinet. A similar one is available in JK's Interiors Patch Collection -- you choose.

Compatibility -- please read before posting
This mod only checks whether the battles of Solitude or Windhelm are completed; no ties, edits, or relation to the Civil War at all.

The quest's start condition is: GetStageDone(CWFinale [QUST:000D1444], 500) = 1
Is the vanilla CWFinale quest -- the one which cleans up the final siege and the CW questline -- in stage 500?
Yes --> Start the rebuilding quest.
No --> Wait.

Q: What happens if a mod prevents the CWFinale quest from going to stage 500?
A: This waits forever and does absolutely nothing.

This said, you know your mods, I don't -- please try to decide yourselves whether 'any-name-here' is compatible or not.