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, the one ending with the death of Tullius or Ulfric.

You'll be notified 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 give a donation, the box is removed from the temple.
If you offer some gold, the box stays, but locked (is there a key somewhere?... yes). After 24 hours the gold will be removed from the box.

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
Shamelessly steal the donations -- Get your gold back plus 3,286 Septims. Repair in about 30/31 days.

A note...
I imagined the weary hero walking to the city gates and riding away into the sunset.
That means that the quest only starts when you're not into one of the five hold capitals, if you mainly fast-travel from city to city it might be quite delayed.
Sorry if it's inconvenient, but I guess that anyone will have a walk outside, sooner or later.

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

Patches

• The patch for Immersive Citizens allows Heimskr to go back to his rebuilt house, and that's the only change it does. 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 (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 if it's being kept up to date though.

Compatibility
It should be compatible with almost everything, apart from mods that have the same scope (post-war repairs) and, possibly, mods that change the Civil War final battle.
To clarify, this mod only checks whether the battle of Solitude (or Windhelm) is completed, no ties, edits, or relation to the Civil War at all.
If the mod 'name-here' allows for those battles to take place and complete, the mod is compatible.
If the mod 'another-name-here' doesn't allow for those battles to take place and complete, the mod is not compatible.
This said, you know your mods, I don't, please decide yourselves whether the mod 'any-name-here' is compatible or not.