This is basically a request to integrate some of the functionality of File (Field) Paths into core. Here's the file field configuration page, once the module is enabled:

The redirect and active/retractive update bit can be left for contrib (or to be solved by #854); the transliteration bit is already implemented in Backdrop (admin/config/media/file-system); the redirect can be left for a separate issue (or contrib); what I find a valuable feature though is the ability to use tokens for file names and file paths. The later is what this issue here is about.
Original request
There is a feature request for this filed in the File (Field) Paths module issue queue that is marked as "closed/fixed". It states that Entity API exposes this as a token (I think it's the Entity Token sub-module actually):

Looking at our token browser though, the deepest I could get to is [node:field_image].
Recent comments
Sorry - it is the field_group module. However, last night I switched off proxying to the site and proved to Cloudflare Support that it was NOT a Backdrop issue. After a thorough...
Cloudflare specific advice needed [SOLVED]
@ian I am getting lost what module are you using that is version 1.x-1.6.0 and 1.x-1.7.0 ?
Cloudflare specific advice needed [SOLVED]
I'm also now getting a 403 when trying to access admin/reports/updates - it seems to be Backdrop generated but there is nothing in the Recent Log messages other than my login as admin...
Cloudflare specific advice needed [SOLVED]