Wordpress - Remove update nags for non-admins

In wp-admin/includes/update.php file

if ( current_user_can('update_core') )
        $msg = sprintf( __('An automated WordPress update has failed to complete - <a href="%s">please attempt the update again now</a>.'), 'update-core.php' );
    else
        $msg = __('An automated WordPress update has failed to complete! Please notify the site administrator.');

We can see that messages are different based on the current user role and this is maintenance_nag.

Basically we have two update nags and can be found in admin-filters.php

add_action( 'admin_notices', 'update_nag',      3  );
add_action( 'admin_notices', 'maintenance_nag', 10 );

So to remove second message we can use(also check for current user role if you want this only for non-admins)

remove_action( 'admin_notices', 'maintenance_nag', 10 );

For multi-site use

remove_action( 'network_admin_notices', 'maintenance_nag', 10 );

@bravokeyl is the probably the best answer to your immediate problem.

But to address the following:

Is there a way to hook into an action or filter and remove ALL the update nag messages for non-admin users?

No. Nag messages in WordPress are just a callback to added to the admin_notices hook which print some HTML to the page. They are practically the same as error or success messages, or any other 'notice' from WordPress or any other plug-in or theme for that matter.

Hiding the nags via CSS is hacky. It's also liable to some false positives as some plugins/themes will, incorrectly, use the .update-nag class to provide the desired styling to their own notices.

A much less hacky way is to explicitly remove each callback that you don't want printing notices (for non-admins). But this comes at a (probably very low cost) of maintaining that list and ensuring there are no notices that 'slip the net'.

here is complete code, which seems to work at this moment:

add_action('admin_head', function() {
    if(!current_user_can('manage_options')){
        remove_action( 'admin_notices', 'update_nag',      3  );
        remove_action( 'admin_notices', 'maintenance_nag', 10 );
    }
});