Modpack version
1.52 server upon updating directly to a 1.56 server

Can the issue be reproduced?
Unknown but if you wanna try, set up a server from version 1.52, complete quests on it, then update to 1.56 and look at the result.

Logs
Server log won't paste anywhere so I will just include the file itself.
latest.log

Issue
I followed the update instructions for the server and, as I feared, something went horribly wrong with the quests.
When I updated I had to use the default load command to get any quests at all, ontop of that everything was reset. We're all the way to Draconic Evolution in the pack so manually activating and claiming the quests to that point is seriously a last resort.

Try running the quest load commands, these used to be recommended to run after every update, I'm not sure why they are not any more:

/bqs_loot default load
/bq_admin default load

Booting up the game now to try that

It doesn't appear to have worked

QuestProgress is stored in the betterquesting folder of your world. Your backups have that file as well, just use that :)