Hey! I also ran into this problem. If you open a template without installing any of the fonts you should get a pop-up in Publisher about missing fonts, which should lead you to the following messages in the font manager. 
The missing fonts are ITC Serif Gothic, AndroclesOpti-Heave and Futura PT. You can fix this as follows.
I'm guessing the absence of the latter two fonts went under watt's radar because these were already installed on their machine. I'll check in with watt.
Have you tried opening, editing, and saving another file in Affinity? It sounds like you might be having issues with your Affinity installation (I’ve read some recommendations to reinstall affinity from their website using the .exe extension but have not experienced this issue myself.
For missing files, you’ll want to move everything to the same folder (before opening Publisher), open the template, open Resource Manager in the windows drop down, and then update or replace the assets using the new file location (the folder where you moved the assets using).
If you can open and work with other Affinity Publisher docs, you could always copy and paste template content over and save there.
hope this helps!
watt
Thanks so much for your response, Watt. I’ve tried all of the advice I’ve found, but so far the only thing that’s working is your copy and paste advice haha. Here’s hoping I’m able to make it through without it getting corrupted again. I suppose if it does, I can just copy and paste again to extend the timer. Thanks again!
