After much research, this appears to be bugged. I tested multiple regions and thoroughly switched my entire system formatting to the UK (locale, keyboard, timezone). I clicked all the buttons, entered all the admin prompts, logged out and rebooted to test system changes, and still couldn't get the indicator to display in dmy format ("10 Sep"). (I had the same issue with the greeter too, for what it's worth.)
The desired setting for this is certainly
Settings > Language & Region.
- "Language" appears to successfully set the system language (and thus have the date in dmy in other languages that use that style)
- If "Region" has an effect, it is not readily apparent, and
- "Formats" visibly changes a date format shown grayed out on the screen. It would follow that changing the format from United States (mdy) to United Kingdom (dmy) that the menu indicator/greeter would change to match the change in the grayed out formatting (see image))
But alas, it does not work as intended. I filed these bugs:
- Indicator's displayed date/time does not respect formatting set in Switchboard Locale Plug (Language & Region) (Related: Use same time format for greeter and indicator)
- No feedback after clicking "Set System Language"
Please subscribe to the bug reports so that the devs can ask for help testing solutions. If you want to give the open source devs some extra incentive, try putting a developer bounty on the task. I'll report back when the bug is cleared.