The below information can be useful for debugging in the following cases:
1. No emails seem to appear in email queue section Promotions > Abandoned Cart Email > Emails Queue.
2. Emails are not being sent automatically despite the settings on the schedule tab Promotions > Abandoned Cart Email > Rules > Rule Name > Schedule Tab.
3. Magento cron is misconfigured.
Debug mode works for registered users only so you will need to create an account for an email address that was never used before.
Please enable debug mode at System > Configuration > Amasty Extensions > Abandoned Cart Email:
Specify the domain name without @sybmol from which the cart would be abandoned, for example, gmail.com (this implies that extension will only trigger the emails that are ended on gmail.com domain unless debug mode is disabled).
Create a simple abandoned cart rule without conditions, add a few records to the schedule:
Add a product to cart, go to the checkout page and leave the tab opened.
Go to emails queue section in the backend (in ~10 secs) and refresh the page.
The emails should appear in the queue section and further be sent out (disregard the schedule in the rule).
If you received the emails on the specified email address the module logic is correct and the reason for not-sending emails is more likely to lie in cron misconfiguration.
Here you can learn more about Magento cron and how it should be configured.