Thank you for your post!
Global Fields are session based when hosted, therefore you wouldn't be able to update the Global Field for ALL clients via one of the client machines. There is another suggestion of setting it via startup script in Setting global fields in a FileMaker Server session.
I hope this helps and I'm going to move this thread from the FileMaker Community Feedback Space, which is specifically for input on the Community itself, to the Discussions Space where you should receive even more views and potentially more advice!
We set global fields and variables via start up script all the time via the method TSPigeon briefly mentions.
We have a script set up in File | File Options via the the OnFirstWindowOpen trigger to run a script.
The script set's a global field with the company name. It set's another global field with the company logo. From a record with nonglobal fields in the database. Should the company name or logo ever change, we just go to the fields in that one record and edit the company name and/or logo as needed. The next time a user opens the file, the updated name or logo will appear in those global fields.