TwentyTwelve child theme mobile menu not working

I had exactly the same problem. In the base TwentyTwelve theme there has been a code change so that the site navigation changed from h3 to button.

You need to update header.php file in the child theme to use

<button class="menu-toggle"><?php _e( 'Menu', 'twentytwelve' ); ?></button>

See WordPress TRAC #28824 comment 13

The problem occurs when you upgrade the base theme from v1.4 to v1.5.
It’s not restricted to mobile devices – you can recreate the problem with a window width less than 600 pixels.