I am using MailMate but BusyContacts is not showing any emails

You need to enable Indexed for Messages under MailMate Settings

NOTE: BusyContacts now supports showing Mail in macOS 10.15.

BusyContacts uses Spotlight to find and list email messages from email clients. MailMate is currently one of the apps that is compatible with BusyContacts post 10.14 (Apple has disabled integration with Apple Mail spotlight in recent macOS updates).

In order to be able to find these emails, you need to first set MailMate as the default email client:

Launch Apple Mail. Open Settings.
Select "MailMate" from the "Default Email Reader" drop down

Launch MailMate and Open MailMate Settings
Under General > Messages Folder, check "Indexed for Finder / Spotlight"
Check the 'Custom Location' option and use the default option of saving the index under your home directory

Quit BusyContacts and relaunch.