I knew that one of my clients web sites is running an older version of WordPress. When I upgrade WordPress I like to know exactly what version is running so I can use something like WinMerge to compare every single file in the WordPress installation to verify specifically what is going to be upgraded. This is useful if you run into any problems you can attempt to only roll back specific files to see if it resolves the WordPress upgrade issue.
There could be a couple issues causing WordPress to not send mail however in a server move the most likely is that the previous server was using Sendmail and the new server is using a different mail server like Postfix or vice versa. There also are a couple ways to resolve the issue and I will explain each method below however I recommend installing a WordPress Add-on to resolve the issue versus modifying a core WordPress file that could inadvertently be overwritten in the future.
The two ways which are described in detail to resolve this issue are to either install the WP-Mail-SMTP WordPress plugin or to modify the pluggable.php file located in the /wp-root/wp-includes directory. Each method will change the way that the $phpmailer function sends mail.