IOS – Physical keyboard layout menu not showing when keyboard connected to iPhone / iPad (iOS 13/14)

Although it isn't exactly what you asked for, one possible solution is to use the built-in ability to rapidly switch Input Sources (aka keyboard layouts). To make this function useable, you'll need to do a little setup.

Open System Preferences: Keyboard and select the Keyboard Pane. Check the option to Show Keyboard & Character Viewers in menu bar (1), then click on the Input Sources… button (2).

The Language and Text preference pane will open to the Input Sources tab. Select the input sources (3) you want to access on a regular basis. Fewer is better, remember you can always return to this tab and change your selections. When you've made your choices click on the Keyboard Shortcuts… button (4).

The Keyboard preferences pane will open to the Keyboard Shortcuts tab. Select Keyboard and Text Input from the column on the left (5). Then check the options to enable the keyboard shortcuts to navigate between input sources. (6).

If you stay with the default shortcuts, they will conflict with the default Spotlight shortcuts. Select Spotlight from the column on the left, and either deactivate or change the keyboard shortcut assigned to the Spotlight functions.

Now when you need to type a special character, use the hotkey to select the appropriate Input Source and use Lion's long hold function to bring up the choices for each character.

I am currently using Romanian as my Input Source . I know this because it's displayed in the Keyboard and Character Viewers menu item. A long press on a provides:

Once I've typed the Romanian character, I can use the keyboard shortcut to return to the U.S. input source.

When I tried enabling the setting (Input source options: Allow a different one for each document), new windows seemed to just use the current input source.

Anyway, you could try running sudo /System/Library/CoreServices/Setup\ Assistant.app/Contents/MacOS/Setup\ Assistant and selecting US as the default input method. (You have to go through the steps of creating a new account, but you can delete it later.) It modifies at least /var/root/Library/Preferences/ByHost/com.apple.HIToolbox.*.plist.

If that doesn't work, try running sudo languagesetup and selecting English, or sudo defaults write /Library/Preferences/.GlobalPreferences.plist AppleLocale en_US and restarting.