Always look at the simple things first
Recently I was asked to take a look at SPD workflow on development system that was not sending emails.
Normally this works fine as it uses the standard emailing features of the SharePoint Platform, as long as the outgoing email settings are configured.
In this instance the outgoing email settings were correct for the environment in question, so a quick test I created an alert on a list, normally this would send an email straight away saying that an alert has been created but no email arrived.
If you have access to the server console the first test is to make sure you can ping your SMTP relay server, this will be the server you have referenced in your outgoing SMTP server configured in Central Administration.
If you can ping the server try using TelNet to connect on Port 25, if telnet times out and fails to connect then you probably have a firewall issue.
In this instance we were getting:
Which was an indication that the development server was not allowed to relay email via the SMTP server, as soon as we had the development server added to the allowed SMTP relay list we could connect via TelNet and send alerts and emails from SharePoint and development continued.
So always check the simple things 1st !
SharePoint (and Project Server) Shenanigans 