I have the same issue even after rebooting my computer. Update - I tried saving to all the normal folders on the Mac all failed, my external drive failed but the save succeeded with Dropbox and iCloud.
When using Save As the following message happens "Failed to save document: /Users/.../file name.afphoto. Save Failed. Access was denied. [ACCESS] Do you wish to save the document to a new file? macOS Mojave 10.14.5, Affinity Photo 1.7.0 I get the same "failed to save" response when selecting save to a new file.