Hey there,
When you say your site is down, how do you mean?
White screen?
Some error message?
Something else?
I see you mentioned that you went to update themes after the WP update, so the rest of the admin and site was fine after that core update?
If you could please check your sites error_log, you could also enable WP_DEBUG, to output the WordPress debugging information you can open your wp-config.php file and change:
define('WP_DEBUG', false);
To:
define('WP_DEBUG', true);
You can also have these errors output to a debug.log file located in your /wp-content/ folder, to do this add the following to your wp-config.php file:
define('WP_DEBUG_LOG', true);
Using WP_DEBUG would put the errors on the site, you can hide those and just have them out put the WP_DEBUG_LOG (if set to true) by using this as well:
define('WP_DEBUG_DISPLAY', false);
Let me know what you find.