I've recently upgraded to Window 10, and ever since then my backtick is not working. It's a keyboard with a US layout, and in my language settings this seems to be properly set up. When I switch to another layout (I used German for testing), the (physical) button is working. So the keyboard itself is fine.
Here are some pictures for reference:
Settings
Keyboard
Best Answer
Update (2018-07-05): Recent versions of Windows 10 may have changed the location of the settings menu. See the answer of @Wingu for an alternative.
I finally figured it out. The problem is that the Windows language switcher uses the tilde to switch between layouts. To fix this, go to Advanced Language Settings in the Control Panel:
Click on "Change Language Bar Hotkeys", which opens up the following dialog:
Here, click on "Change Key Sequence", and on the following dialog change the assignment away from the tilde/grave accent key. Or simply unassign it.