Shouldn't we reuse the ZANSHIN_OVERRIDE_DATETIME environment variable here? Just to avoid situations where currentDate() and currentDateTime() are out of sync because the two variables aren't properly set?

Maybe once the transition is done we can switch to a single ZANSHIN_OVERRIDE_DATE variable indeed, but that feels a bit early here.

I don't want to risk timezone issues with datetimes resolving to a different day, and that would require another commit to convert from DATETIME to DATE everywhere.
If you prefer I can discard this and bundle the DATETIME=>DATE change into the upcoming big commit (which is about QDateTime=>QDate change everywhere)