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

The quest starts soon after the final siege and battle, usually when you leave Solitude or Windhelm.
You'll be notified (not by the shrewd courier who managed to plant a GPS tracker in your undies... 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 subsidize and speed-up the reconstruction, or ignore it and spend your septims elsewhere.
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 gets locked (is there a key somewhere?... yes), and 24 hours later all the gold will be removed from the box.

All the cities will be repaired and reverted to before-siege status: Solitude/Windhelm and Whiterun.

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

Quest start
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 can only start when you're not in one of the five hold capitals. For example, if you mainly fast-travel from city to city, it might be quite delayed.
More technically, the quest is started by a "change location" event, provided that the new location is in the Tamriel worldspace and that the CWFinale quest has gone to stage 500 (after the victory speech by Tullius in Windhelm or Ulfric in Solitude.)

Quest completion
The quest completes when you enter an interior cell after the required period has expired.
This is to prevent the possible oddity of seeing swapping buildings and disappearing stuff.

Optional 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. You can get a similar one (if you don't trust mine) in JK's Interiors Patch Collection.
  

Globals In case you wish to customize the repair timers.
No_Gold   (Default 492 game hours  --  Approx. 20/21 days)
Five_K_Gold   (336  --  15/16)
Ten_K_Gold   (216  --  10/11)
Fifteen_K_Gold   (96  --  5/6)
Steal_Gold   (720  --  30/31)
Change the values only before the start of the quest.
The game-hour values don't match the actual days, they are approximations.
There is some fixed (script) and variable (how soon you interact with the box) time added.
There is also the randomness of how soon you enter an interior cell once the timer has expired (if you spend months just sitting beside your campfire and hunting rabbits.....) 
Anyway, by adding (or subtracting) ten hours, you get exactly ten more (or less) hours for the timer.
If you're lazy: Extended Repair Times.

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
Meaning: Is the vanilla CWFinale quest (the one which cleans up the final siege and the CW questline after the victory speech by Tullius or Ulfric) in stage 500?
Yes ---> The rebuilding quest can start.
No ---> Wait.
If a mod prevents the CWFinale quest from going to stage 500, this waits forever and does absolutely nothing.

You probably know what your mods do, I don't.
If you don't know, please ask the authors.