If there is an issue sending an email message then it will be logged to the application event log as a SMTP error. When Nintex sends an email it does so using the standard Microsoft API. If there is no error within the actual workflow or the SharePoint logs/application event log then the issue is due to the email routing and most likely an issue with your email server.
Firstly, check if the configured SMTP server address is the same as the one configured for SharePoint?
We would recommend that you temporarily install an SMTP service on one of your servers and point Nintex Workflow to that (in Nintex Workflow Management > Global Settings).
Then when Nintex Workflow sends an email, it will go into a drop folder and you can confirm that Nintex Workflow is doing what it is expected to do with regards to sending an email to the configured SMTP service. If you find that emails arrive in the drop folder, then Nintex Workflow is successfully sending emails, and the issue is most likely something to do with your “real” SMTP service.
One thing to note also is that, the ReplyTo address in the global settings page, must match the lazy approval settings email, and when user hits Reply in Outlook for LA email, it will have the global Replyto email.
I hope this helps!