We use a twentytwelve child theme on our site and it too stopped working on mobile devices recently (within the past 2 weeks) (Menu button present but not working). I banged my head against the problem for a while and finally figured it out:
We have a custom header.php that was a copy of the stock one with code added for our SSL certificate but with no changes to the navigation. We haven't changed it in a while (sometime in June I think). Our header.php, and the original from twenty twelve until very recently used an <h3> tag for the Menu button. The newest version of the theme uses <button> instead. The code that updates the toggle is in navigation.js, which we did not touch in our child theme, but it too was updated to look for a button object instead of an <h3> like it used to.
I figured it out by running the site in the Chrome debugger and saw navigation.js look for a button object that wasn't there. I then checked the theme files and compared to a backup from July and discovered the change.
Anyway, I fixed it by changing the <h3> to <button> in my child theme and everything works again.
It appears that there was a theme update on or about 9/8/14 that introduced the problem. I will pay closer attention to theme updates from now on.