First, you need to determine what exactly is happening on your website:
1. Does your current sending method work?
If you are using PHP Mail or Sendmail as your sending method (Settings page > Send With tab), try this: The Alternate Method to Verify if PHP Mail Works (“Your own website”)
If you are using Gmail to send your newsletters (which we do not recommend): Sending With Gmail Doesn’t Work… What To Do?
In these cases, we always suggest you to use a SMTP Provider, read here why.
2. Are you receiving an error when trying to send a test email?
Is the error Could not instantiate mail function? If it is, read this page.
If the error message says something with the word “JSON” on it, try to increase your server memory to 128M with this plugin: Change Memory Limit
3. Are your newsletters not sent on time?
Are you experiencing delays of newsletters being sent?
For example, automatic newsletter to be sent immediately or your scheduled newsletter was sent late?
It’s normal to experience a delay of a few minutes because of how WordPress manages its task scheduler (also known as the Cron).
Read more about that here: Understanding the Cron