Recently, I upgraded EE v2.9 website to EE v5.3.2 which is multi-language website managed with Publisher. I am facing problem after upgrade. None of custom fields like Textarea, RTE etc populating content.
The issue with fields not populating will be hard to diagnose without you providing more information about how the fields were defined and how you are trying to access them / include in templates.
As for the Publisher / NavEE support - wasn't too hard to find this on the BoldMinded site, which is indicated as working with EE4/5.
If this doesn't work with EE5, I'd get in touch with BoldMinded and see what they suggest.