Modified Programmer's Dvorak Layout with German Umlauts

X11 Version

I created a slight modification of the original programmer's Dvorak keyboard scheme for X11, with the following changes: Quick install: download https://immanuel-albrecht.de/pdvrkde/usr-share-X11-xkb-symbols-us and place it under /usr/share/X11/xkb/symbols/us, and choose the English programmer's Dvorak keyboard layout. This might work for you:
curl https://immanuel-albrecht.de/pdvrkde/usr-share-X11-xkb-symbols-us |\
     sudo tee /usr/share/X11/xkb/symbols/us > /dev/null

Windows Version

Download this file, unzip it, and install via setup.exe.

macOS Big Sur Version

In order to install the default Programmer's Dvorak layout I use home brew:
brew install --cask programmer-dvorak
Then I apply a small patch to it:
curl https://immanuel-albrecht.de/pdvrkde/umlauts.diff |\
     sudo patch /Library/Keyboard\ Layouts/Programmer\ Dvorak.bundle/Contents/Resources/Programmer\ Dvorak.keylayout
Optionally, you might want to apply the numpad patch, too. (Since MacBooks do not come with number pads, I usually do not bother to do this.)
curl https://immanuel-albrecht.de/pdvrkde/numpad-patch.diff |\
     sudo patch /Library/Keyboard\ Layouts/Programmer\ Dvorak.bundle/Contents/Resources/Programmer\ Dvorak.keylayout
That's it, you're all set to enjoy Programmer's Dvorak with German umlauts. (Please note that some office applications use Shift+Alt+A as a shortcut, you have to disable this shortcut in order to write Ä within these applications!)