After clicking through various posts, I have found a solution!

There are two shortcuts that may work on your Mac: Cmd + Shift + < and Cmd + Shift + >

I said "may work", because the active keyboard layout plays a role:

With the US layout active, only Cmd + Shift + < works, and the same is true for Russian Phonetic and Devanagari Phonetic, which have the < and > characters on the same keys as the US layout.

But with the Spanish, German, or French keyboard layouts, the < and > characters are on a different key (the top left key) and here both shortcuts work.

In practice, this doesn't make much difference, as few people will have more than 2 or 3 windows open. As long as you know where to find the < key on your keyboard layout, you're probably OK.

I still wonder why Cmd + tilde doesn't work for me, but I won't lose sleep over it… My MacOS is pretty old (El Capitán) so that may be a factor.

In what position is the tilde key or your keyboard because if you have a special (non en-US) keyboard (layout) then the tilde might be in a different position (scancode) ?

That's a valid question, as I'm normally typing on a compact keyboard from Packard Bell, and using non-Apple hardware can cause weird results.

So, to test my results, I plugged in an Apple keyboard – and got the same result there: Typing Cmd-Shift-` doesn't switch to a different Firefox window.

In the attached screenshot you can see the tilde displayed on the top left key, because the Shift key is held down. You can also see that the active keyboard layout is "U.S." What I can't show you is that Firefox doesn't switch windows :-)

Actually, my guess is that it's not the key's scancode that's important, but the actual character normally produced by the key. After all, the thing worked for me with < both on the U.S. layout (with < on the comma key) and on other-language layouts (with < on the top left key).