How to fix your time of day being out of sync
Set iPresentInterval in SkyrimPrefs.ini to 1
Load the save, use the wait command and observe what hour the game will change to a new day, mine was sometime within 7PM.
Once you know within what hour the day change occurs, use the wait command to get as close as possible before the day changes (I waited until 5:48PM, another hour and the day changed). Without using the wait command, sit and watch time go by until you know what exact minute the borked day change takes place (for me it was 6:44PM).
Once you know what exact time it happens at, convert that time to a 24 hour clock. So I changed 6:44PM to 18:44.
Now for some maths. See how far from 00:00 the next day the time updates. So mine was 5 hours, 16 minutes early (24 - 18.44).
Get the number of minutes you have, in my case 16, and multiply that by 0.6. I got 9.6.
Replace the old value you had for minutes with the new value, taking the decimal point out. So I had 5.16 before, my new value is now 5.96 (replacing .16 with the new .96).
Open the console and type "Set GameHour to GameHour + VALUE", where value is the number you just worked out. In my case, 5.96.
This will get your day to update normally at exactly 12AM.