Missing keyboard shortcut in El Capitan 10.11.5 to switch between windows within app
The Keyboard item in System Preferences has a Shortcuts tab. The Keyboard item in the Shortcuts tab has one item labeled Move focus to next window. The binding for that has a  `   rather than a  ~   (with the  Cmd ⌘  key; I don't know how to type a cloverleaf), but either binding has the same effect for me (that box is checked on my Mac).

It's possible you simply need to check that box; in any case, that seems like the first place to look.