Opening any of the iWork suite brings up over 80 dialogs in a row
I managed to fix this, I hope this helps anyone else facing a similar problem.

1. Installing the fonts.

Font Book v9.0

Greyed out fonts that need to be installed have a "Download" button but if you look at the information there's a download link in there. I found that by copying and pasting it somewhere else the URL had a double slash "//" which was probably interfering with the ability to download it. It's actually a link to fonts in the /System directory.

I found most of the fonts I needed here:

/System/Library/Frameworks/ApplicationServices.framework/Frameworks/ATS.framework/Support/FontSubsets


By right clicking the font name and using the "Add Fonts…" dialog I could browse to the font and install them.

2. Permissions

The apps still didn't work. One of the Numbers crash reports said that /Applications/Numbers.app/Contents/SharedSupport/DocumentResources/Metadata.plist couldn't be accessed. When I checked all the permissions were screwed up - some folders were owned by root, others had permissions that locked out anyone but the owner. So:

sudo chmod -R g+rwx "/Applications/Pages.app/Contents/"
sudo chmod -R g+rwx "/Applications/Numbers.app/Contents/"
sudo chmod -R g+rwx "/Applications/Keynote.app/Contents/"


and chown commands across the right dirs seems to have fixed that.

Can't say I'm impressed with this turn of events, the standards at Apple really have been slipping badly of late.